For every natural number n we introduce a new weak choice principle $\mathrm {nRC_{fin}}$:

Given any infinite set x, there is an infinite subset $y\subseteq x$ and a selection function f that chooses an n-element subset from every finite $z\subseteq y$ containing at least n elements.

By constructing new permutation models built on a set of atoms obtained as Fraïssé limits, we will study the relation of $\mathrm {nRC_{fin}}$ to the weak choice principles $\mathrm {RC_m}$ (that has already been studied in [3] and [6]):

Given any infinite set x, there is an infinite subset $y\subseteq x$ with a choice function f on the family of all m-element subsets of y.

Moreover, we prove a stronger analogue of the results in [6] when we study the relation between $\mathrm {nRC_{fin}}$ and $\mathrm {kC_{fin}^-}$ which is defined by

Given any infinite family $\mathcal {F}$ of finite sets of cardinality greater than k, there is an infinite subfamily $\mathcal {A}\subseteq \mathcal {F}$ with a selection function f that chooses a k-element subset from each $A\in \mathcal {A}$.

For every $n\in \omega \setminus \{0,1\}$ we introduce the following weak choice principle:

Moreover, we consider the following choice principle:

We will discuss the relations between these two choice principles and their relations to other well-known weak choice principles. Moreover, we will discuss what happens when we replace $\mathcal {F}$ by a linearly ordered or a well-ordered family.

We show that MRP + MA implies that ITP(λ,ω2) holds for all cardinal λ ≥ ω2. This generalizes a result by Weiβ who showed that PFA implies that ITP(λ, ω2) holds for all cardinal λ ≥ ω2. Consequently any of the known methods to prove MRP + MA consistent relative to some large cardinal hypothesis requires the existence of a strongly compact cardinal. Moreover if one wants to force MRP + MA with a proper forcing, it requires at least a supercompact cardinal. We also study the relationship between MRP and some weak versions of square. We show that MRP implies the failure of □(λ, ω) for all λ ≥ ω2 and we give a direct proof that MRP + MA implies the failure of □(λ, ω1) for all λ ≥ ω2.