1 The original conjecture

Throughout the paper $\kappa $ will denote a regular uncountable cardinal, and a cardinal greater than $\kappa $ . We let denote the collection of all subsets of of size less than $\kappa $ , and the noncofinal ideal on . An ideal J on is fine if . For further definitions see the end of this section.

According to Jech [Reference Jech18, p. 409], one of the key concepts in the theory of large cardinals is saturation of ideals. The starting point of our story is Solovay’s seminal result [Reference Solovay58] that for any value of $\kappa $ , the nonstationary ideal on $\kappa $ is nowhere weakly $\kappa $ -saturated. There are no large cardinals involved, so how does this fit with Jech’s statement? The first remark to make is that the proof of Solovay’s result needs choice. In fact, by another result of Solovay, under the Axiom of Determinacy, the nonstationary ideal on $\omega _1$ is prime (i.e., weakly $2$ -saturated). Returning to ZFC, we have Solovay’s result at one end, and measurable cardinals with their normal measures at the other. And in between? For each cardinal $\rho $ between $2$ and $\kappa $ , set-theorists carefully determined the exact consistency strength of $\kappa $ carrying a (normal or at least) $\kappa $ -complete, weakly $\rho $ -saturated ideal.

In the glorious early days of the study of (which was seen as a two-cardinal generalization of $\kappa $ ), it was systematically attempted to establish versions of known results on $\kappa $ (see, for instance, the problems listed in Section 0 of [Reference Kunen and Pelletier24]). Failures were no problem, since they were seen as productive. By analyzing what went wrong in the attempted generalization, one acquired a better understanding of and its specificity (for example, see the work of Solovay, Menas [Reference Menas47], and Kunen and Pelletier [Reference Kunen and Pelletier24] on normal measures on without the partition property). Thus Menas [Reference Menas46] boldly conjectured that holds for any possible values of $\kappa $ and , where asserts that the nonstationary ideal on is nowhere weakly -saturated.

Progress on this two-cardinal version of Solovay’s splitting result was initially slow. As observed by Kanamori in [Reference Kanamori21], most results were about with replaced by . For example, an early result of Jech [Reference Jech17] stated that if $\kappa $ is a successor and regular, then is nowhere weakly -saturated. This was later improved by Baumgartner who proved that for any possible values of $\kappa $ and , is nowhere weakly -saturated.

The conjecture was finally refuted by Baumgartner and Taylor [Reference Baumgartner and Taylor3] who showed the consistency of the failure of $MC_1 (\omega _1, \omega _2)$ . Their result can be revisited as follows. We put ( $=$ the least size of any subset of ) not in ).

The conjecture somehow survived its refutation by Baumgartner and Taylor, as the so-called splitting problem, the general problem of computing the degree of weak saturation of ideals on . In how many stationary pieces can this or that stationary subset of be partitioned? Given a cardinal , what is the consistency strength of the existence of a (normal or maybe just) $\kappa $ -complete, weakly $\rho $ -saturated, fine ideal on ?

Recall that for a fine ideal J on , Jensen’s diamond principle asserts the existence of $s_x$ for such that $Z_B = \{x : B \cap x = s_x \}$ lies in $J^+$ for all . Since $\{x \in Z_a : a \subseteq x\} \cap \{x \in Z_b : b \subseteq x\} = \emptyset $ for any two distinct members $a, b$ of , implies that J is not weakly -saturated. In [Reference Matsubara43] Matsubara established the consistency of the conjecture by proving that it holds in L. As observed by Shioya [Reference Shioya56], one way to proceed would have been to appeal to the fact that in L, ( and hence) holds for all . Instead, Matsubara relied on the result of Baumgartner in the case when , and completed his proof by showing that follows from . A new proof of this result can be found in [Reference Matsubara and Shioya45]. Let us recall that an ideal J on is precipitous if for all generic , the ultrapower is well-founded. By a result of Foreman [Reference Foreman10], any countably complete -saturated ideal on is precipitous. As is well-known, any normal, fine, weakly -saturated ideal on is -saturated (in fact -saturated) and hence precipitous. In [Reference Matsubara and Shioya45] Matsubara and Shioya establish that no countably complete ideal J on with $cof (J) = non (J)$ is precipitous. It follows that if , then no restriction of is precipitous (and hence holds). Let us observe that by pushing the counting argument used in [Reference Matsubara43], one obtains the following.

Since (see [Reference Matet, Péan and Shelah40] for the exact value of ), it follows that if , then ( holds for every , and hence) holds.

If is a strong limit cardinal with , then , so Matsubara’s result shows that for any value of $\kappa $ , there are always many values of (of cofinality less than $\kappa $ ) for which holds. Further, Matsubara and Shelah [Reference Matsubara and Shelah44] (see also [Reference Matet and Shelah41]) have shown that if is a strong limit cardinal with , then no restriction of is precipitous (and hence holds). To sum up the results of this section, may consistently fail. On the other hand, we have the following.

A large cardinal is needed to obtain situations when [Reference Donder, Koepke and Levinski8] (see also [Reference Kanamori21, p. 345]). Thus Fact 1.4 shows that if GCH holds and there are no large cardinals in an inner model, then holds. However the large cardinals in question are of a modest size, since Donder, Koepke, and Levinski [Reference Donder, Koepke and Levinski8] showed that if and is $\kappa $ -Erdős, then the set is stationary. We can do better than this. Recall that any normal, fine, weakly -saturated ideal on is precipitous. Hence if GCH holds, $\kappa $ is weakly inaccessible, , and there is no normal, precipitous, fine ideal on , then no normal, fine ideal on is weakly -saturated. Now if carries a precipitous ideal, then by a result of Magidor (see [Reference Matsubara43]), there is a cardinal $\sigma $ of Mitchell order $\sigma ^{++}$ in some inner model. Thus the following formulation seems preferable.

This discussion is continued in Section 9 where we show that if there are no large cardinals in an inner model, then Menas’s conjecture is equivalent to a weak form of GCH.

Let us at last provide the missing definitions. By an ideal on an infinite set X, we mean a nonempty collection J of subsets of X such that (a) $X \notin J$ , (b) $P(A)\subseteq J$ for all $A \in J$ , (c) $A \cup B \in J$ whenever $A, B \in J$ , and (d) $\{x\} \in J$ for all $x \in X$ . Given an ideal J on X, we let $J^+ = P(X) \setminus J$ , $J^{\ast } = \{ A \subseteq X : X \setminus A \in J\}$ , and $J \vert A = \{ B \subseteq X : B \cap A\in J\}$ for each $A \in J^+$ . For a cardinal $\rho ,J$ is $\rho $ -complete if $\bigcup Q \in J$ for every $Q \subseteq J$ with $\vert Q \vert < \rho $ . We let $non (J) =$ the least size of a set in $J^+$ . $cof (J)$ denotes the least cardinality of any $Q \subseteq J$ such that $J = \bigcup _{A \in Q} P(A)$ . Given an infinite set Y and $f : X \rightarrow Y$ , we let $f (J) = \{ B \subseteq Y : f^{-1} (B) \in J\}$ . For a cardinal $\sigma $ , J is weakly $\sigma $ -saturated (respectively, $\sigma $ -saturated) if there is no $Q \subseteq J^+$ with $\vert Q \vert = \sigma $ such that $A \cap B = \emptyset $ (respectively, $A \cap B \in J$ ) for any two distinct members $A, B$ of Q. J is nowhere weakly $\sigma $ -saturated if for any $A \in J^+$ , $J \vert A$ is not weakly $\sigma $ -saturated.

For a regular uncountable cardinal $\tau $ , $I_{\tau }$ (respectively, $NS_{\tau }$ ) denotes the noncofinal (respectively, nonstationary) ideal on $\tau $ . For each regular cardinal $\mu < \tau $ , $E^{\tau }_{\mu }$ , (respectively, $E^{\tau }_{< \mu }$ ) denotes the set of all infinite limit ordinals $\alpha < \tau $ such that $\mathrm {cf}(\alpha ) = \mu $ (respectively, $\mathrm {cf}(\alpha ) < \mu $ ).

2 Second and third versions of the conjecture

In view of the Baumgartner–Taylor result, it is tempting to repair the conjecture by replacing with the (weaker) assertion that is nowhere weakly -saturated (notice that since , and are equivalent for ). This would be more in line with Solovay’s result, which after all does not assert that $NS_{\kappa }$ is nowhere $\kappa ^{<\kappa }$ -saturated (this holds if and only if $2^{<\kappa } = \kappa $ ), but only that it is nowhere $cof (I_{\kappa })$ -saturated. However, Gitik [Reference Gitik12] proved that it is consistent relative to a large cardinal that “ $\kappa $ is inaccessible and $NS_{\kappa , \kappa ^+} \vert W$ is weakly $\kappa ^+$ -saturated for some W (and hence $MC_2 (\kappa , \kappa ^+)$ fails).” Krueger [Reference Krueger23] showed that one could take $W = \{x : o.t (x) = \vert x \cap \kappa \vert ^+ \}$ , and Shioya [Reference Shioya56] obtained a new proof of Gitik’s result starting from the hypothesis that $\kappa $ is $\kappa ^+$ -supercompact. Magidor (see [Reference Di Prisco and Marek7]) had shown that for all values of $\kappa $ and , is nowhere weakly $\kappa $ -saturated, which is thus optimal. Thus Solovay’s result that $NS_{\kappa }$ is nowhere weakly $\kappa $ -saturated does generalize, but the generalization only asserts that is nowhere weakly $\kappa $ -saturated.

Notice that if $\kappa $ is -Shelah and , then, as shown by Johnson [Reference Johnson20], the set $W = \{x : o.t. (x) = \vert x \cap \sigma \vert ^+ \}$ lies in (and hence in ), but, by another result of Johnson [Reference Johnson19], is not weakly -saturated. Thus the stationarity of $W = \{x : o.t. (x) = \vert x \cap \kappa \vert ^+ \}$ does not guarantee by itself that $NS_{\kappa , \kappa ^+} \vert W$ is weakly $\kappa ^+$ -saturated.

By Gitik’s result, is still too strong. One way to weaken it would be to require only that is nowhere weakly -saturated for some S. Now Usuba [Reference Usuba60] established that if , then there is a stationary subset S of with the property that no normal, fine ideal J on with $S \in J^+$ is weakly -saturated. Taking our cue from this, we let assert the existence of S in such that no normal, fine ideal J on with $S \in J^+$ is weakly -saturated. Then any normal extension of will be nowhere weakly -saturated. Clearly, the larger S is, the better.

3 Pcf theory

In this section we start working on using tools of Shelah’s pcf theory.

3.1 Pseudo-Kurepa families

For two cardinals $\sigma $ and $\pi $ , asserts the existence of with $\vert X \vert = \pi $ such that $\vert X \cap P (b) \vert < \kappa $ for all . It is simple to see that ( , and hence) holds. These families can be used to strengthen the result of Baumgartner mentioned above. First the case when $\kappa $ is a successor cardinal:

Fact 3.1 [Reference Matet26].

Suppose that $\kappa $ is a successor cardinal and holds. Then no $\kappa $ -complete, fine ideal on is weakly $\pi $ -saturated.

Proof The proof is an easy modification of that of Baumgartner. Let J be a $\kappa $ -complete, fine ideal on . Put $\kappa = \nu ^+$ . We can assume that $\pi $ is greater than or equal to $\kappa $ (since we have seen that ( and hence) ) always holds) and (by Fact 1.6(i)) regular. Select with $\vert X \vert = \pi $ such that for all . For , pick a one-to-one function $f_b : X \cap P (b) \rightarrow \nu $ . By $\kappa $ -completeness and fineness of J, for each $x \in X$ , we may find $S_x \in J^+ \cap P (\{b : x \subseteq b\})$ and $\gamma _x < \nu $ such that $f_b (x) = \gamma _x$ for all $b \in S_x$ . There must be $\gamma < \nu $ and $W \subseteq X$ with $\vert W \vert = \pi $ such that $\gamma _x = \gamma $ for any $x \in W$ . Then clearly $S_x \cap S_y = \emptyset $ for any two distinct members $x, y$ of W.

In particular, as observed by Matsubara [Reference Matsubara42] long ago, if $\kappa $ is a successor cardinal, then no $\kappa $ -complete, fine ideal on is weakly -saturated.

We use more in the case when $\kappa $ is weakly inaccessible. Given an ideal J on and a cardinal $\tau $ with , J is $\tau $ -normal if for any $A \in J^+$ and any $f : A \rightarrow \tau $ such that $f(a) \in a$ for all $a \in A$ , there is $B \in J^+ \cap P(A)$ such that f is constant on B. Notice that -normality is the same as normality.

Fact 3.2 [Reference Matet28].

Suppose that $\tau $ and $\pi $ are two cardinals with and $\tau < \pi $ , and J is a $\tau $ -normal, fine ideal on . Suppose further that there is with $\vert X \vert = \pi $ such that . Then J is not weakly $\pi $ -saturated.

Note that if X is as in the statement of the fact, then $\vert X \cap P (c) \vert < \kappa $ for all (so X witnesses that holds). Further note that it follows from Fact 3.2 that if $\kappa $ is weakly inaccessible, then no $\kappa $ -normal, fine ideal J on with is weakly -saturated. Thus by combining Facts 3.1 and 3.2, we obtain the following.

Proposition 3.3. If , then holds.

Let us return to the special case of Fact 3.2 when and . If $\kappa $ is weakly inaccessible, , and J is a normal, fine, weakly -saturated ideal on , then it gives us that the set of all such that ( $\vert b \vert> \vert b \cap \tau \vert $ for every cardinal $\tau $ with , and hence) lies in $J^{\ast }$ . In the special case when for some regular cardinal $\sigma < \kappa $ , we can even conclude that , since it is simple to see that for any cardinal $\chi $ with , .

For another remark, letting C denote the set of all

such that (a) $o.t. (b)$ is an infinite limit ordinal, and (b) $b \setminus \tau \not = \emptyset $ , then clearly

, and moreover

for all $b \in C$ with

. For normal ideals there is the following related result of Usuba (see the proof of Proposition 6.1 in [Reference Usuba59]) who proved it using generic embeddings.

Fact 3.4. Suppose that is regular, and let J be a normal, fine ideal on such that $S \in J^+$ , where S is the set of all such that ( $o.t. (b)$ is an infinite limit ordinal with) $\mathrm {cf} (o.t. (b)) < o.t. (b)$ . Then J is not weakly -saturated.

Proof For each infinite limit ordinal $\delta $ , select an increasing function $t_{\delta } : \mathrm {cf} (\delta ) \rightarrow \delta $ such that $\sup (ran (t_{\delta })) = \delta $ . For $b \in S$ , let $g_b : o.t. (b) \rightarrow b$ enumerate b in increasing order, and set $\tau _b = \mathrm {cf} (o.t. (b))$ and $\xi _b = g_b (\tau _b)$ . By normality of J, we may find $Y \in J^+ \cap P (S)$ and such that $\xi _b = \xi $ for all $b \in Y$ . For $\alpha \in b \in Y$ , put:

• • $F (\alpha , b) =$ the least $\zeta < \tau _b$ such that ;

• • $\beta ^b_{\alpha } = g_b (t_{o.t. (b)} (F (\alpha , b)))$ ;

• • $\gamma ^b_{\alpha } = g_b (F (\alpha , b))$ .

Thus $F (\alpha , b) < \tau _b$ , $\beta ^b_{\alpha } \in b$ and $\gamma ^b_{\alpha } \in b \cap \xi $ . For , there must be, by normality of J, $T_{\alpha } \in J^+ \cap P (\{b \in Y : \alpha \in b\})$ , , and $\gamma _{\alpha } < \xi $ such that for all $b \in T_{\alpha }$ , $\beta ^b_{\alpha } = \beta _{\alpha }$ and $\gamma ^b_{\alpha } = \gamma _{\alpha }$ . Since is regular, we may find and $\gamma < \xi $ such that $\gamma _{\alpha } = \gamma $ for all $\alpha \in d$ . Finally, pick with the property that $\beta _{\alpha } < \alpha '$ whenever $\alpha < \alpha '$ are in e. Given $\alpha < \alpha '$ in e, we claim that $T_{\alpha } \cap T_{\alpha '} = \emptyset $ . Suppose otherwise, and pick $b \in T_{\alpha } \cap T_{\alpha '}$ . Then $F (\alpha , b) = F (\alpha ', b)$ (since

$$ \begin{align*} g_b (F (\alpha, b)) = \gamma^b_{\alpha} = \gamma_{\alpha} = \gamma = \gamma_{\alpha'} = \gamma^b_{\alpha'} = g_b (F (\alpha', b))), \end{align*} $$

and therefore . Contradiction.

3.3 Large pseudo-Kurepa families from scales

Throughout the remainder of this section and Sections 4 and 5 , we let $\theta , \pi , A$ , and I be such that:

• • $\theta $ and $\pi $ are two cardinals such that and $\theta < \pi $ ;

• • A is a set of regular cardinals smaller than $\theta $ such that $\vert A \vert < \kappa $ and $\sup A = \theta $ ;

• • I is an ideal on A with $\{A \cap a : a \in A \} \subseteq I$ ;

• • $\pi = \mathrm {tcf} (\prod A /I )$ .

Further let be an increasing, cofinal sequence in $(\prod A, <_I)$ .

Let $\Phi $ denote a one-to-one onto function from $On \times On$ to $On$ such that $\Phi " (\sigma \times \sigma ) = \sigma $ for any infinite cardinal $\sigma $ . We let be defined by: $y_{\alpha }$ equals $\{\alpha \}$ if $\alpha < \theta $ , and $\{\Phi (a, f_{\alpha } (a)) : a \in A \}$ otherwise.

Observation 3.7. Let $\delta> \theta $ be a good point for $\vec f$ , and let e be a cofinal subset of $\delta $ of order-type $\mathrm {cf} (\delta )$ . Then $\vert \bigcup _{\alpha \in e} y_{\alpha } \vert = \max (\vert A \vert , \mathrm {cf} (\delta ))$ .

Proof By Fact 3.5(ii), we may find a subset X of $e \setminus \theta $ of order-type $\mathrm {cf} (\delta )$ such that the sequence is strongly increasing. Pick $Z_{\xi } \in I$ for $\xi \in X$ so that $f_{\beta } (a) < f_{\xi } (a)$ whenever $\beta < \xi $ are in X and $a \in A \setminus (Z_{\beta } \cup Z_{\xi })$ . Now select $k \in \prod _{\xi \in X} (A \setminus Z_{\xi })$ , and put $t = \{\Phi (k (\xi ), f_{\xi } (k (\xi ))) : \xi \in X \}$ . It is simple to see that $\vert t \vert = \vert X \vert = \mathrm {cf} (\delta )$ . It follows that

$$ \begin{align*} \vert \bigcup_{\alpha \in e} y_{\alpha} \vert \geq \vert \bigcup_{\alpha \in X} y_{\alpha} \vert \geq \max (\vert A \vert, \mathrm{cf} (\delta)). \end{align*} $$

On the other hand, it is readily verified that .

Fact 3.8.

1. (i) [Reference Matet37] There is a closed unbounded subset $C_{\vec {f}}$ of $\pi $ , consisting of infinite limit ordinals, with the property that any $\delta $ in $C_{\vec {f}}$ satisfying one of the following conditions, where $\rho $ denotes the largest limit cardinal less than or equal to $\mathrm {cf} (\delta )$ , is a good point for $\vec f$ :

   1. (a) $(\max (\rho , \vert A \vert ))^{+3} < \mathrm {cf} (\delta )$ .

   2. (b) $\rho ^{\vert A \vert } < \mathrm {cf} (\delta )$ .

   3. (c) $\vert A \vert < \mathrm {cf} (\rho )$ .

   4. (d) $\vert A \vert < \rho $ and I is $(\mathrm {cf} (\rho ))^+$ -complete.

   5. (e) and $\mathrm {pp} (\rho ) < \mathrm {cf} (\delta )$ .

2. (ii) [Reference Matet28] Suppose that and there is a closed unbounded subset C of $\pi $ such that every $\delta \in C$ of cofinality $\kappa $ is a good point for ${\vec f}$ . Then holds, as witnessed by $X = \{y_{\alpha } : \alpha < \pi \}$ .

Thus, for instance, ${\mathcal A}_{\omega _4,\sigma } (\omega _1, \sigma ^+)$ holds for every uncountable cardinal $\sigma $ of cofinality $\omega $ . On the other hand by a result of Todorcevic (see [Reference Matet28]), if , for every cardinal $\tau $ with , and , then fails.

The scale is good if there is a closed unbounded subset C of $\pi $ with the property that every infinite limit ordinal $\delta $ in C such that $\vert A \vert < \mathrm {cf} (\delta ) <\sup A$ is a good point for $\vec f$ .

If $\pi $ is the successor of a cardinal $\tau $ (possibly greater than $\sup A$ ) at which the weak square principle $\square _{\tau }^{\ast }$ holds, then [Reference Cummings, Foreman and Magidor6, Reference Matet31] the scale $\vec f$ is good. Let us recall that the failure of $\square _{\tau }^{\ast }$ for singular $\tau $ has, in the words of Jech [Reference Jech18, p. 702] (see also [Reference Cummings5]), the consistency strength of (roughly) at least one Woodin cardinal.

3.5 The isomorphism method

One way to show that an ideal J on is not weakly $\pi $ -saturated is to establish that it is isomorphic to some ideal K on $P_{\kappa } (\pi )$ that is itself not weakly $\pi $ -saturated. In this subsection we consider some situations when this can (or cannot) be done.

For $i = 1, 2$ , let $X_i$ be an infinite set, and let $K_i$ be an ideal on $X_i$ . We say that $K_1$ is isomorphic to $K_2$ if there are $W_1 \in K_1^{\ast }$ , $W_2 \in K_2^{\ast }$ , and a bijection $k : W_1 \rightarrow W_2$ such that $K_1^{\ast } = \{D \subseteq W_1 : k"D \in K_2^{\ast }\}$ .

Notice that $K_1$ is isomorphic to $K_2$ if and only if there are $W_1 \in K_1^{\ast }$ , $W_2 \in K_2^{\ast }$ , and a bijection $k : W_1 \rightarrow W_2$ such that (a) $\{k"D : D \in K_1^{\ast } \cap P (W_1)\} \subseteq K_2^{\ast }$ , and (b) $\{k^{- 1} (B) : B \in K_2^{\ast } \cap P (W_2)\} \subseteq K_1^{\ast }$ . It easily follows that $K_1$ is isomorphic to $K_2$ just in case $K_2$ is isomorphic to $K_1$ .

For a cardinal $\sigma> \kappa $ and a $\kappa $ -complete ideal G on $P_{\kappa } (\sigma )$ , $\overline {\mathrm {cof}} (G)$ denotes the least cardinality of any $Q \subseteq G$ such that for any $B \in G$ , there is $Z \in P_{\kappa } (Q)$ with $B \subseteq \bigcup Z$ .

Observation 3.11. For $i = 0, 1$ , let $\sigma _i$ be a cardinal greater than $\kappa $ , and let $K_i$ be a fine ideal on $P_{\kappa } (\sigma _i)$ . Suppose that $K_1$ is $\kappa $ -complete, and $K_1$ and $K_2$ are isomorphic. Then $K_2$ is $\kappa $ -complete, and moreover $\overline {\mathrm {cof}} (K_1) = \overline {\mathrm {cof}} (K_2)$ .

Fact 3.12 [Reference Matet36].

The following are equivalent:

1. (i) $f (K_1) = K_2$ for some one-to-one $f : X_1 \rightarrow X_2$ .

2. (ii) There are $W_2 \in K_2^{\ast }$ and a bijection $k : X_1 \rightarrow W_2$ such that $K_1^{\ast } = \{D \subseteq W_1 : k"D \in K_2^{\ast }\}$ .

We use the isomorphism method to give new versions of Facts 3.1 and 3.2. Let us start with the situation when $\kappa $ is a successor cardinal. There is a more informative proof of Fact 3.1 which runs as follows. For the first case, when , proceed as in the original proof. Now for the second case, suppose that $\kappa $ is a successor cardinal, J is a $\kappa $ -complete, fine ideal on , , and X is a size $\pi $ subset of with the property that $\vert X \cap P (b) \vert < \kappa $ for all . Letting be a one-to-one enumeration of X, define by $f (b) = b \cup \{\alpha : x_{\alpha } \subseteq b \}$ . Clearly, f is one-to-one, so by Fact 3.12, J and $f (J)$ are isomorphic. Since by the first case, $f (J)$ is not weakly $\pi $ -saturated, J is not weakly $\pi $ -saturated either. It is worth stressing that, as the following shows, there are situations when the condition on the existence of an X as above cannot be dispensed with.

Observation 3.13. Suppose that $\kappa $ is a successor cardinal, and some $\kappa $ -complete, fine ideal J on with is isomorphic to some fine ideal K on $P_{\kappa } (\pi )$ . Then holds.

Proof By Observation 3.11, K is $\kappa $ -complete, and moreover . By Proposition 5.7 in [Reference Matet, Péan and Shelah39], the desired conclusion follows.

Observation 3.13 can be applied with . More interestingly, suppose that is a strong limit cardinal of cofinality less than $\kappa $ . Then by a result of Shelah [Reference Shelah53], for some , so and moreover since is nowhere -saturated by Fact 1.6(ii), so is . Thus if GCH holds, $\kappa $ is a successor cardinal, , and fails, we cannot use the isomorphism trick to show that is not weakly saturated.

Let us now turn to the case when $\kappa $ is weakly inaccessible.

Observation 3.14. Suppose that , $\kappa $ is weakly inaccessible, $\sigma $ is a cardinal with , and J is a $\sigma $ -normal, fine ideal on . Suppose further that there exist a closed unbounded subset C of $\pi $ and $D \in J^+$ such that $\delta $ is a good point for $\vec {f}$ whenever $\delta \in C \cap E^{\pi }_{\vert b \cap \sigma \vert ^+}$ for some $b \in D$ . Then J is not weakly $\pi $ -saturated.

Proof Recall that is defined by: $y_{\alpha }$ equals $\{\alpha \}$ if $\alpha < \theta $ , and $\{\Phi (a, f_{\alpha } (a)) : a \in A \}$ otherwise. Define by

$$ \begin{align*} g (b) = (b \cap \sigma) \cup \{\alpha \in C \setminus \sigma : y_{\alpha} \subseteq b\}. \end{align*} $$

Notice that $g (b) \cap \theta = b \cap \theta $ . It follows that g is one-to-one in case .

Claim. Let $b \in D$ . Then $\vert g (b) \vert = \vert b \cap \sigma \vert $ .

Proof of the claim

Suppose otherwise. Pick a subset e of $g (b) \setminus \sigma $ of order-type $\vert b \cap \sigma \vert ^+$ , and put $\delta = \sup e$ . Then clearly, $\delta \in C \cap E^{\pi }_{\vert b \cap \sigma \vert ^+}$ , and therefore $\delta $ is a good point for $\vec {f}$ . But then by Observation 3.7,

$$ \begin{align*} \vert \bigcup_{\alpha \in e} y_{\alpha} \vert \geq \mathrm{cf} (\delta)> \vert b \cap \sigma \vert \geq \vert b \cap \theta \vert. \end{align*} $$

This is a contradiction, since $\bigcup _{\alpha \in e} y_{\alpha } \subseteq b \cap \theta $ , which completes the proof of the claim.

Set $K = g (J \vert D)$ and $Z = g" (D)$ . Then, as is readily checked, K is a $\sigma $ -normal, fine ideal on $P_{\kappa } (\pi )$ , and moreover $Z \in K^{\ast }$ . By the claim, $\vert x \vert $ = $\vert x \cap \sigma \vert $ for all $x \in Z$ , so by Fact 3.2, K is not weakly $\pi $ -saturated, and hence neither is J.

Corollary 3.15. Suppose that , $\kappa $ is weakly inaccessible, $\sigma $ is a cardinal with , and J is a weakly $\pi $ -saturated, $\sigma $ -normal, fine ideal on .Then $\Upsilon \in J^{\ast }$ , where $\Upsilon $ denotes the set of all for which there are stationarily many $\delta < \pi $ such that (a) $\delta $ is not a good point for ${\vec f}$ , and (b) $\mathrm {cf} (\delta ) = \vert b \cap \sigma \vert ^+$ .

Proof Suppose otherwise, and let . Let T be the set of all $\delta < \pi $ such that either $\delta \notin E^{\pi }_{\vert b \cap \sigma \vert ^+}$ for every $b \in D$ , or $\delta $ is a good point for $\vec {f}$ . Then $\pi \setminus T$ must be stationary. Hence we may find $\mu < \pi $ , and a stationary subset W of $\pi \setminus T$ , such that (a) $\mu = \vert b \cap \sigma \vert ^+$ for some $b \in D$ , and (b) $W \subseteq E^{\pi }_{\mu }$ . Notice that for any $\delta \in W$ , $\delta $ is not a good point for $\vec {f}$ . Thus $b \in \Upsilon $ . Contradiction.

The corollary extends several known results, including one by Shelah (see [Reference Eisworth9, Theorem 4.63]) asserting that above a supercompact $\chi $ , there is no good scale at any cardinal $\rho $ of cofinality smaller than $\chi $ , and a more detailed result of Sinapova [Reference Sinapova57, Lemma 8] for the special case when $\rho < \chi ^{+ \chi }$ .

4 The function $\psi $

Throughout the section it is assumed that . We need a new method to handle situations when the results established so far do not apply. In this quest the function $\psi $ considered in this section will play a key role.

Define $\varphi : P_{\theta } (\pi ) \rightarrow \prod A$ by: $\varphi (w) (a)$ equals $\sup (w \cap a)$ if $\sup (w \cap a) < a$ , and $0$ otherwise. Further define $\psi : P_{\theta } (\pi ) \rightarrow \pi $ by $\psi (w) =$ the least $\alpha $ such that .

To show that an ideal K on is not weakly $\pi $ -saturated, it will suffice to prove that the ideal on $\pi $ is (included in some ideal on $\pi $ that is) not weakly $\pi $ -saturated. As a first step, in this section we compare $\psi \vert P_{\kappa } (\pi )$ and the sup function on $P_{\kappa } (\pi )$ . The main results are Observation 4.7, which will be used in Section 5, and Observation 4.3.

A subset C of $P_{\kappa } (\pi )$ is strongly closed if $\bigcup X \in C$ for all $X \in P_{\kappa }(C) \setminus \{\emptyset \}$ . $SNS_{\kappa , \pi }$ denotes the collection of all $B \subseteq P_{\kappa }(\pi )$ such that $B \cap C = \emptyset $ for some strongly closed C in $I^+_{\kappa , \pi }$ . It is easy to see that $SNS_{\kappa , \pi }$ is a $\kappa $ -complete, fine ideal on $P_{\kappa } (\pi )$ . Furthermore, $SNS_{\kappa , \pi } \subset NS_{\kappa , \pi }$ .

Let $\mu $ be a regular cardinal less than $\kappa $ . An ideal K on $P_{\kappa } (\pi )$ is $(\mu , \kappa )$ -normal if for any $G \in K^+$ and any $k : G \rightarrow P_{\mu } (\pi )$ with the property that $k (x) \subseteq x$ for every $x \in G$ , there is d in $P_{\kappa } (\pi )$ and $H \in K^+ \cap P (G)$ such that $f (x) \subseteq d$ for all $x \in H$ . We let denote the smallest fine, $\kappa $ -complete, $(\mu , \kappa )$ -normal ideal on . As observed in [Reference Matet33], the set of all $x \in P_{\kappa } (\pi )$ such that $\sup e \in x$ for all $e \in P_{\mu } (x)$ lies in $NS_{\mu , \kappa , \pi }^{\ast }$ . Furthermore, the set of all $x \in P_{\kappa } (\pi )$ such that $\sup x$ is a limit ordinal of cofinality $\mu $ lies in $NS_{\mu , \kappa , \pi }^+$ .

A subset C of $P_{\kappa } (\pi )$ is $\mu $ -closed if $\bigcup _{i < \mu } c_i \in C$ for every increasing sequence in $(C, \subset )$ . A subset D of $P_{\kappa } (\pi )$ is a $\mu $ -club if it is a $\mu $ -closed, cofinal subset of $P_{\kappa } (\pi )$ . We let $N\mu $ - $S_{\kappa ,\pi }$ be the set of all $H \subseteq P_{\kappa } (\pi )$ such that $H \cap D = \emptyset $ for some $\mu $ -club $D \subseteq P_{\kappa } (\pi )$ . Let $Y_{\mu }^{\kappa , \pi }$ denote the set of all nonempty $x \in P_{\kappa }(\pi )$ such that:

• • For any $\alpha \in x$ , $\alpha + 1 \in x$ .

• • For any regular infinite cardinal $\sigma \in \kappa \setminus \{\mu \}$ , and any increasing sequence of elements of x, $\sup \{\beta _{\xi } : \xi < \sigma \} \in x$ .

Notice that for any $x \in Y_{\mu }^{\kappa , \pi }$ and any cardinal of cofinality greater than or equal to $\kappa $ , $\sup (x \cap \tau )$ is a limit ordinal of cofinality $\mu $ . It is remarked in [Reference Matet30] that $Y^{\kappa , \pi }_{\mu }$ and the set $\{x \in P_{\kappa } (\pi ):\vert x\vert = \vert x \cap \kappa \vert \geq \mu \}$ are both $\mu $ -clubs.

Observation 4.3 is of no use in the (critical) case when $\vert A \vert = \mu $ , which must be handled separately. In what follows we present Shelah’s approach, which is based on Namba combinatorics (that is, properties of trees of height $\omega $ ). So from here on to Observation 4.7 we suppose that ( $\mathrm {cf} (\theta ) = \omega $ and) $\vert A \vert = \aleph _0$ . Let be a one-to-one enumeration of A, and set ${\cal I} = u (I)$ , where $u : A \rightarrow \omega $ is defined by: $u (\theta _i) = i$ .

An ideal H on $\omega $ is a P-point if for any $F : \omega \rightarrow H$ , there is $G \in H^{\ast }$ such that $G \cap F (j)$ is finite for all $j < \omega $ .

Assume that ${\cal I}$ is a P-point (i.e., that given $B_n \in I$ for $n < \omega $ , there is $C \in I^{\ast }$ that meets each $B_n$ in a finite set). For $\alpha < \pi $ , define $F_{\alpha } \in \prod _{i < \omega } \theta _i$ by $F_{\alpha } (i) = f_{\alpha } (\theta _i)$ .

${\cal T}_{\pi }$ denotes the collection of all nonempty $T\subseteq \bigcup _{n < \omega } {}^n\pi $ such that for any $t \in T$ , $\{ t \vert n : n < \mathrm {dom} (t) \} \subseteq T$ and $\vert \{\alpha < \pi : t \cup \{(\mathrm {dom} (t), \alpha )\} \in T \} \vert = \pi $ .

Let $T \in {\mathcal T}_{\pi }$ . Set $[T] = \{ f\in {}^{\omega }\pi : \forall n < \omega (f \vert n \in T)\}$ . For $t \in T$ , put $T\ast t = \{ s \in T : s \subseteq t$ or $t \subseteq s\}$ . $[T]$ is endowed with the topology obtained by taking as basic open sets the members of the family $\{ [T \ast t] : t \in T \}$ .

Let $\sigma $ be a cardinal greater than $\kappa $ , and let $\mu $ be a regular cardinal less than $\kappa $ . For $Q \subseteq P_{\kappa } (\sigma )$ , $G_{\kappa ,\sigma }^{\mu } (Q)$ denotes the following two-person game consisting of $\mu $ moves. At step $\alpha < \mu $ , player I selects $a_{\alpha } \in P_{\kappa } (\sigma )$ , and II replies by playing $b_{\alpha } \in P_{\kappa } (\sigma )$ . The players must follow the rule that for $\beta < \alpha < \mu $ , $b_{\beta }\subseteq a_{\alpha } \subseteq b_{\alpha }$ . II wins if and only if $\bigcup _{\alpha < \mu } a_{\alpha }\in Q$ .

$NG_{\kappa ,\sigma }^{\mu }$ denotes the collection of all $Q \subseteq P_{\kappa } (\sigma )$ such that II has a winning strategy in $G_{\kappa ,\sigma }^{\mu } (P_{\kappa } (\sigma ) \setminus Q)$ .

The following generalizes a result of Shelah [Reference Shelah54] (in which $\pi = \theta ^+$ and ${\cal I} = I_{\omega }$ ).

Lemma 4.6. Let $S \in NS_{\pi }^+ \cap P (E_{\omega }^{\pi })$ . Then

$$ \begin{align*} \{ b \in P_{\omega_1} (\pi) : \sup b \in S \text{ and } \psi (b) = \sup b )\} \in NS_{\omega_1, \pi}^+. \end{align*} $$

Proof Fix a closed unbounded subset C of $P_{\omega _1} (\pi )$ . Pick

so that:

• • For $h : \omega \rightarrow \pi $ , $k (h) = \bigcup _{n < \omega } k (h \vert n)$ .

• • For $t : m + 1 \rightarrow \pi $ , $\{ t (m) \} \cup k (t \vert m) \subseteq k (t)$ .

Clearly, if $h \in {}^{\omega }\pi $ , $i \in \omega $ , and $\beta \in \theta _i$ are such that $\varphi (k(h)) (\theta _i)> \beta $ , then $\varphi (k(h \vert m)) (\theta _i)> \beta $ for some m with . Hence, using Fact 4.4, we may construct inductively $T \in {\cal T}_{\pi }$ and $\chi : T \rightarrow \theta $ such that for any $t \in T$ and any $h \in [T\ast t]$ , .

We define a strategy $\tau $ for player II in $G_{\kappa ,\pi }^{\omega } (P_{\kappa } (\pi ))$ as follows. Consider a play of the game where I’s successive moves are $d_0, d_1, \dots $ . Using her strategy $\tau $ , II will successively play $e_0, e_1, \dots $ so that:

• • $d_j \cup \{ \psi (d_j) \} \subseteq e_j$ .

• • $\sup e_j$ is an infinite limit ordinal greater than $\sup d_j$ .

• • For any $t \in T$ with $\mathrm {ran} (t) \subseteq e_j$ , $\chi (t) \in e_j$ , and moreover

  $$ \begin{align*} \sup \{ \gamma \in e_j : t \cup \{ (\mathrm{dom} (t), \gamma)\} \in T \} = \sup e_j. \end{align*} $$

Now $\{ b \in P_{\omega _1} (\pi ) : \sup b \in S \} \in NS_{\omega _1, \pi }^+$ , so by Fact 4.5(iv), $\tau $ is not a winning strategy for II in $G_{\kappa ,\pi }^{\omega } (\{ b \in P_{\omega _1} (\pi ) : \sup b \notin S \})$ . Hence there must be a play of this game where I’s successive moves are $d_0, d_1, \dots $ , II successively plays $e_0, e_1, \dots $ using her strategy $\tau $ , and I wins. Thus $\sup e \in S$ , where $e = \bigcup _{j < \omega } e_j$ . Put $\delta = \sup e$ . Notice that for any $j < \omega $ , . It follows that , since $\psi (d_{j + 1}) \in e_{j + 1}$ . Define $\unicode{x0142} :\omega \rightarrow {\cal I }$ by $\unicode{x0142} (j) = \{ i < \omega : \varphi (e_j) (\theta _i)> F_{\delta } (i) \}$ . By P-pointness of ${\cal I}$ , there must be $G \in {\cal I}^{\ast }$ such that $G \cap \unicode{x0142} (j)$ is finite for all $j < \omega $ . Pick an increasing sequence $< n_j : j < \omega>$ of elements of $\omega $ so that whenever $j < \omega $ and $i \in G \setminus n_j$ . Now construct $h \in [T]$ so that $\{h (i) : i < n_0\} \subseteq e_0$ , and for any $j < \omega $ :

• • $h(i) \in e_j$ whenever ;

• • $h(n_{j + 1})> \sup (e_j)$ .

Put $b = k (h)$ . Then clearly, $b \in C$ , and moreover $\mathrm {ran} (h) \subseteq b$ . It follows that .

Claim. Let $j < \omega $ and $i \in G$ with . Then .

Proof of the claim

Since $\mathrm {ran} (h \vert i) \subseteq e_j$ , $\chi (h \vert i) \in e_j \cap \theta _i$ . Hence, , which completes the proof of the claim.

It follows from the claim that . Thus, , so $\sup b = \psi (b)$ .

Observation 4.7. Let $S \in NS_{\pi }^+ \cap P (E_{\omega }^{\pi })$ . Then

$$ \begin{align*} \{ w \in P_{\kappa} (\pi) : \sup w \in S \text{ and } \psi (w)= \sup w\} \in (NG^{\omega}_{\kappa,\pi})^+. \end{align*} $$

Proof For $\kappa = \omega _1$ , this is just Lemma 4.6. Let us now assume that $\kappa> \omega _1$ . Set

$$ \begin{align*} Q = \{ x \in P_{\omega_1} (\pi^{< \kappa}) : \sup (x \cap \pi) \in S \text{ and } \psi (x \cap \pi) = \sup (x \cap \pi )\}. \end{align*} $$

Then by Fact 4.5(iii) and (iv) and Lemma 4.6, $Q \in NS_{\omega _1,\pi ^{< \kappa }}^+$ . By Fact 4.5(v), we may find $y : \pi ^{< \kappa } \rightarrow P_{\kappa } (\pi )$ such that (a) for any $\beta \in \pi $ , $\beta \in y (\beta )$ , and (b) $NG^{\omega }_{\kappa ,\pi } = {\overline y} (NS_{\omega _1, \pi ^{< \kappa }})$ , where ${\overline y} : P_{\omega _1} ( \pi ^{< \kappa } )\rightarrow P_{\kappa } (\pi )$ is defined by ${\overline y} (x) = \bigcup _{\delta \in x} y(\delta )$ . Note that $x \cap \pi \subseteq {\overline y} (x)$ for all $x \in P_{\omega _1} (\pi ^{< \kappa })$ . Put $c = \{ i < \omega : \theta _i \geq \kappa \}$ . Note that $c \in {\cal I}^{\ast }$ . Let C denote the set of all infinite $x \in P_{\omega _1} (\pi ^{< \kappa })$ such that:

• • For any $\eta \in x$ and any $i \in c$ , there is $\beta \in x \cap \theta _i$ with $\sup (y (\eta ) \cap \theta _i) < \beta $ .

• • For any $\eta \in x$ , there is $\beta \in x \cap \pi $ with $\sup (y (\eta )) < \beta $ .

Then clearly, C is a closed unbounded subset of $P_{\omega _1} (\pi ^{< \kappa })$ . Furthermore for any $x \in C$ , $\psi (x \cap \pi ) = \psi ({\overline y} (x))$ and $\sup (x \cap \pi ) = \sup ({\overline y} (x))$ . Thus ${\overline y}" (Q \cap C)$ lies in $(NG^{\omega }_{\kappa ,\pi })^+$ , and moreover it is included in the set $\{ w \in P_{\kappa } (\pi ) : \sup w \in S$ and $\psi (w)= \sup w\}$ .

For any $S \in (NS_{\pi } \vert E_{\omega }^{\pi })^+$ , $\{ w \in P_{\kappa } (\pi ) : \psi (w) \in S\} \in (NG^{\omega }_{\kappa ,\pi })^+$ by Observation 4.7, and therefore by Fact 4.5(iii). Thus . Let us observe that it is not necessarily the case that $\psi " W \in NS_{\pi }^+$ for all . To see this, recall that the bounding number $\mathfrak {b}_{\pi }$ denotes the least cardinality of any $F \subseteq {}^{\pi } \pi $ with the property that there is no $g \in {}^{\pi } \pi $ such that $\vert \{\beta < \pi : f (\beta ) \geq g (\beta \} \vert < \pi $ for all $f \in F$ .

Set and . Then , so by Observation 4.9, there must be $D \in NS_{\pi }^{\ast }$ such that . Select T in $I_{\pi }^+ \cap NS_{\pi } \cap P (D \cap E^{\pi }_{\omega })$ , and put . Then clearly, , and moreover $\psi " W \in NS_{\pi }$ .

Another comment is in order. In contrast to Observation 4.3, the conclusion of Observation 4.7 does not assert that the set $\{ w \in P_{\kappa } (\pi ) : \sup w \in S$ and $\psi (w)= \sup w\}$ lies in the dual filter ( $(NG^{\omega }_{\kappa ,\pi })^{\ast }$ ), but only that it does not lie in the ideal ( $(NG^{\omega }_{\kappa ,\pi })$ ). There is a good reason for this. In fact if $\mu = \vert A \vert = \mathrm {cf} (\theta )$ , $\theta ^{< \mu } < \pi $ , and $Z \in NS_{\pi }^+ \cap P(E_{\mu }^{\pi })$ lies in the approachable ideal $I [\pi ]$ , then by a result of [Reference Matet30], for any $\ell : \pi \rightarrow \pi $ ,

$$ \begin{align*} \{ x \in P_{\kappa} (\pi) : \sup x \in Z \text{ and } \psi (x)> \ell (\sup x )\} \in (NG^{\mu}_{\kappa,\pi})^+. \end{align*} $$

On the other hand, we have the following.

Note that if $V = L$ , then by a result of Dieter Donder, for any stationary $Y \subseteq P_{\kappa } (\pi )$ , there exists a stationary $X \subseteq Y$ such that for all $\beta < \pi $ .

Let us finally observe that if $x = ran (f_{\alpha })$ , where $\alpha < \pi $ , then obviously $\psi (x) = \alpha $ . This shows that $\psi (x)$ needs not be a limit ordinal. On the other hand, by Observation 4.7 (respectively, Observation 4.3), if $\vert A \vert = \mu = \omega $ (respectively, $\vert A \vert < \mu $ or I is $\mu ^+$ -complete), the set of all x such that $\psi (x)$ is a limit ordinal of cofinality $\mathrm {cf} (\vert x \cap \theta \vert ) (= \mu )$ lies in $(NG_{\kappa ,\pi }^{\mu })^+$ (respectively, $(NG_{\kappa ,\pi }^{\mu })^{\ast }$ ). One intriguing question which is left unanswered by the results of the next section is whether the set of all x such that $\psi (x)$ is a limit ordinal of cofinality $(\mathrm {cf} (\vert x \cap \theta \vert ))^+$ lies in $J^{\ast }$ for any weakly $\pi $ -saturated, normal, fine ideal J on .

5 The way of Usuba

As in the preceding section, we assume in this one that , and we look for strategies to deal with situations when our ideal J on cannot be shown to be isomorphic to a nice ideal on $P_{\kappa } (\pi )$ . Sticking with ideals, we could attempt to bypass $P_{\kappa } (\pi )$ and associate J with an ideal on $\pi $ of the form $h (J)$ for some . But there are obvious difficulties. One is that $h (J)$ will be $\kappa $ -complete (when we would be more comfortable with $\pi $ -completeness). Another is that if , then J and $h (J)$ will not be isomorphic. So instead we will follow Usuba [Reference Usuba60] whose method consists in moving from J (to its projection on $P_{\kappa } (\theta )$ and then) to an ideal K on $P_{\kappa } (\theta )$ with which is associated a function h taking its values in $\pi $ . The crux of the matter is that if J is weakly $\pi $ -saturated, then so is K.

An ideal on $P_{\kappa } (\sigma )$ , where $\sigma $ is a cardinal greater than or equal to $\theta $ , is adequate if it is $\kappa $ -complete in case $\kappa $ is a successor cardinal, and $\theta $ -normal otherwise.

Let J be an adequate, fine ideal on . We put ${\cal J} = p (J)$ , where is defined by $p (c) = c \cap \theta $ .

We should stress that $\psi $ (or, for that matter, the fact that $\theta $ is singular) does not play any role in the definition of h and K.

Concerning (iv), notice that by $\kappa $ -completeness of K, we can actually simultaneously reflect less than $\kappa $ many stationary sets. By Theorem 2.13 in [Reference Hayut and Lambie-Hanson15], it follows that if J (and hence K) is weakly $\pi $ -saturated, then the square principle $\square (\pi , < \sigma )$ fails for every regular cardinal $\sigma < \kappa $ .

Proof Suppose otherwise. Then the set Q of all $b \in P_{\kappa } (\theta )$ with lies in $K^+$ . Define $r : P_{\kappa } (\theta ) \rightarrow P_{\kappa } (\theta )$ by: $r (b)$ equals b if $\kappa $ is weakly inaccessible, and $\nu $ if $\kappa = \nu ^+$ . For $b \in Q$ , pick $t_b : r (b) \rightarrow h (b)$ such that $\sup (ran (t_b)) = h (b)$ . Let $\chi = \theta $ if $\kappa $ is weakly inaccessible, and $\chi = \nu $ if $\kappa = \nu ^+$ . We define $s : \chi \rightarrow \pi $ , and $Q_{\gamma } \in (K \vert Q)^{\ast } \cap P (Q)$ for $\gamma < \chi $ , as follows. Put $W = \{b \in P_{\kappa } (\theta ) : h (b)> 0\}$ . Notice that $W \in K^{\ast }$ . Given $\gamma < \chi $ , set $W_{\gamma } = W \cap \{b \in P_{\kappa } (\theta ) : \gamma \in b\}$ , and define $g_{\gamma } \in \prod _{b \in W_{\gamma }} h (b)$ by: $g_{\gamma } (b)$ equals $t_b (\gamma )$ if $b \in Q$ , and $0$ otherwise. Since K is weakly $\pi $ -saturated, we may find $\xi _{\gamma } < \pi $ such that $X_{\gamma } \in K^{\ast }$ , where . We now let $s (\gamma ) = \xi _{\gamma }$ . Notice that for all $b \in Q \cap X_{\gamma }$ .

Since K is adequate, the set of all $b \in P_{\kappa } (\theta )$ such that $b \in X_{\gamma }$ for all $\gamma \in b \cap \chi $ lies in $K^{\ast }$ . Hence we may select $b \in Q$ such that (a) $h (b)> \sup (ran (s))$ , and (b) $b \in X_{\gamma }$ for all $\gamma \in r (b)$ . Then

Contradiction.

The ideal ${\cal K} = h (K)$ does have some interesting properties, but it is not clear what can be gained by considering it. By Fact 5.2 and Observations 5.3 and 5.6, the following hold:

• • ${\cal K}$ is a $\kappa $ -complete ideal on $\pi $ extending $NS_{\pi }$ .

• • If K is weakly $\pi $ -saturated, then so is ${\cal K}$ .

• • For any $X \in {\cal K}^+$ and any $g : X \rightarrow \pi $ such that $g (\gamma ) < \gamma $ for all $\gamma \in X$ , there is $\beta < \pi $ with .

• • Suppose that K is weakly $\pi $ -saturated. Then (a) for any $X \in {\cal K}^{\ast }$ and any $g : X \rightarrow \pi $ such that $g (\gamma ) < \gamma $ for all $\gamma \in X$ , there is $\beta < \pi $ with , and (b) the set of all $\gamma \in \pi $ such that $\gamma $ is not a good point for ${\vec f}$ lies in ${\cal K}^{\ast }$ .

A function $g \in \prod A$ is an exact upper bound for some $F \subseteq \{ f_{\alpha } : \alpha < \pi \}$ if (a) for all $f \in F$ , and (b) for any $k \in \prod A$ with $k <_I g$ , there is $f \in F$ with $k <_I f$ .

An infinite limit ordinal $\delta < \pi $ is a more-than-good point for $\vec f$ if there is a cofinal subset X of $\delta $ , and $Z_{\xi } \in I$ for $\xi \in X$ such that $f_{\beta } (a) < f_{\xi } (a)$ whenever $\beta < \xi $ are in X and $a \in A \setminus Z_{\xi }$ . By results of Shelah, for any regular cardinal $\mu $ with $\vert A \vert < \mu < \theta $ , the set of all good points for ${\vec f}$ of cofinality $\mu $ is stationary in $\pi $ . This is also true of more-than-good points.

For $b \in P_{\kappa } (\theta )$ , let $d_b$ be the set of all $\delta \in b$ such that $o.t. (b \cap \delta )$ is a regular cardinal greater than $\vert A \vert $ . For $\delta \in d_b$ , let $G^b_{\delta }$ be the set of all $\beta < h (b)$ with $\mathrm {cf} (\beta ) = o.t. (b \cap \delta )$ with the property that $\beta $ is a good point for . Let $\Omega $ be the set of all $b \in P_{\kappa } (\theta )$ such that $G^b_{\delta }$ is stationary in $h (b)$ for all $\delta \in d_b$ .