Let $p$ be a prime, let $S$ be a non-empty subset of $\mathbb{F}_p$ and let $0\lt \epsilon \leq 1$. We show that there exists a constant $C=C(p, \epsilon )$ such that for every positive integer $k$, whenever $\phi _1, \dots , \phi _k: \mathbb{F}_p^n \rightarrow \mathbb{F}_p$ are linear forms and $E_1, \dots , E_k$ are subsets of $\mathbb{F}_p$, there exist linear forms $\psi _1, \dots , \psi _C\,:\, \mathbb{F}_p^n \rightarrow \mathbb{F}_p$ and subsets $F_1, \dots , F_C$ of $\mathbb{F}_p$ such that the set $U=\{x \in S^n\,:\, \psi _1(x) \in F_1, \dots , \psi _C(x) \in F_C\}$ is contained inside the set $V=\{x \in S^n\,:\, \phi _1(x) \in E_1, \dots , \phi _k(x) \in E_k\}$, and the difference $V \setminus U$ has density at most $\epsilon$ inside $S^n$. We then generalize this result to one where $\phi _1, \dots , \phi _k$ are replaced by homomorphisms $G^n \to H$ for some pair of finite Abelian groups $G$ and $H$, and to another where they are replaced by polynomial maps $\mathbb{F}_p^n \to \mathbb{F}_p$ of small degree.

In this article, we introduce Mallows processes, defined to be continuous-time cadlag processes with Mallows distributed marginals. We show that such processes exist and that they can be restricted to have certain natural properties. In particular, we prove that there exist regular Mallows processes, defined to have their inversion numbers ${\mathrm{Inv}}_j(\sigma )=|\{i\in [j-1] \, : \, \sigma (i)\gt \sigma (\,j)\}|$ be independent increasing stochastic processes with jumps of size $1$. We further show that there exists a unique Markov process which is a regular Mallows process. Finally, we study properties of regular Mallows processes and show various results on the structure of these objects. Among others, we prove that the graph structure related to regular Mallows processes looks like an expanded hypercube where we stacked $k$ hypercubes on the dimension $k\in [n]$; we also prove that the first jumping times of regular Mallows processes converge to a Poisson point process; we eventually provide results regarding the Markov property of the discrete jumping chain corresponding to a Mallows process.

We investigate the asymptotic behaviour of the longest increasing subsequence $L_n$ in a sequence of i.i.d. random variables, for an arbitrary distribution. If the common distribution is atomless then a famous result by Hammersley and Veršik-Kerov states that $L_n$ grows like $2\sqrt {n}$. For any discrete distribution, we characterize the asymptotic order of $L_n$ through a variational problem and provide explicit estimates for classical distributions. The proofs rely on a coupling with an inhomogeneous version of the discrete-time continuous-space Hammersley process. This reveals that, in contrast to the continuous case, the discrete setting exhibits a wide range of growth rates between $\mathcal{O}(1)$ and $o(\sqrt {n})$, depending on the tail behaviour of the distribution. We can then easily deduce the asymptotics of $L_n$ for a completely arbitrary distribution.

In $2023$, Defant and Li defined the Ungarian Markov chain $\mathbf{U}_L$ associated to a finite lattice $L$. This Markov chain has state space $L$, and from any state $x \in L$ transitions to the meet of $\{x\} \cup T$, where $T$ is a randomly selected subset of the elements of $L$ covered by $x$. For any lattice $L$, let $\mathcal{E}(L)$ be the expected number of steps until the maximal element of $L$ transitions into the minimal element in the Ungarian Markov chain. We show that $\mathcal{E}(L)$ is linear in $n$ when $L$ is the weak order on the symmetric group $S_n$, and satisfies an $n^{1-o(1)}$ lower bound when $L$ is the $n^{\text{th}}$ Tamari lattice. This completely resolves a conjecture by Defant and Li and partially resolves another.

We design a Correlation-Decay based quasi-polynomial time deterministic approximation algorithm for computing high dimensional integrals, applied in two settings. The first is computing the volume of an independent set polytope with restrictions. Randomized polynomial time approximation algorithms for computing the volume of a convex body have been known now for several decades, but the corresponding deterministic counterparts are not available, and our algorithm is the first of this kind. The class of polytopes for which our algorithm applies arises as linear programming relaxation of the independent set problem with the additional restriction that each variable takes value in the interval $[0,1-\alpha ]$ for some $\alpha \lt 1/2$. (We note that the $\alpha \ge 1/2$ case is trivial). The method works provided $\alpha \gt 1/2-O(1/\Delta ^2)$, where $\Delta$ is the maximum degree of the graph. When $\Delta =3$ (the sparsest non-trivial case), our method works provided $0.488\lt \alpha \lt 0.5$. Our second setting is to compute the integral of a multi-dimensional separable function, supported by some underlying hyper-graph structure, appropriately defined. Equivalently, our integral is the partition function of a graphical model with continuous potentials. For our method to work, we require that the domain is bounded and the hyper-edge potentials are positive and bounded on the domain. We further assume that the potentials have upper and lower bounds separated by a multiplicative factor of $1 + O(1/\Delta ^2)$, where $\Delta$ is the maximum degree of the graph.

The celebrated canonical Ramsey theorem of Erdős and Rado implies that for $2\leq k\in {\mathbb{N}}$, any colouring of the edges of $K_n$ with $n$ sufficiently large gives a copy of $C_{2k}$ which has one of three canonical colour patterns: monochromatic, rainbow or lexicographic. In this paper we show that if $p=\omega (n^{-1+1/(2k-1)}\log n)$, then ${\mathbf G}(n,p)$ will asymptotically almost surely also have the property that any colouring of its edges induces canonical copies of $C_{2k}$. This determines the threshold for the canonical Ramsey property with respect to even cycles, up to a $\log$ factor.

Previous work of the author and Wolf, and independently Chernikov and Towsner, showed that $3$-uniform hypergraphs of small slicewise VC-dimension admit homogeneous partitions. The results of Chernikov and Towsner did not produce explicit bounds, while the work of the author and Wolf relied on a strong version of hypergraph regularity, and consequently produced a Wowzer type bound on the size of the partition. This paper gives a new proof of this result, yielding $\epsilon$-homogeneous partitions of size at most $2^{2^{\epsilon ^{-K}}}$, where $K$ is a constant depending on the slicewise VC-dimension. This is the first in a series of papers about growth of regular partitions of $3$-uniform hypergraphs. Our main result is a crucial ingredient in the second and third papers in the series, which investigate the growth of two types of regular partitions in hereditary properties of $3$-uniform hypergraphs, namely the weak type first defined by Chung and Haviland–Thomason, and the stronger type first developed by Gowers, and independently, Frankl and Rödl.

Given a (di)graph $H$, we say that a (di)graph $H^{\prime }$ is an $H$-subdivision if $H^{\prime }$ is obtained from $H$ by replacing one or more edges with internally vertex-disjoint paths. Pavez-Signé conjectured that for every $\varepsilon \gt 0$, there exists a constant $C_0\gt 0$ such that for every graph $H$ with $h$ edges and no isolated vertices, if $G$ is a graph on $n\geq C_0h$ vertices and minimum degree $\delta (G)\geq (1+\varepsilon )\frac {n}{2}$, then $G$ contains a spanning $H$-subdivision. This conjecture was later resolved by Lee [European J. Combin. 124 (2025), 104059]. In this paper, we strengthen Lee’s result. Specifically, we prove that for any digraph $D$ on $n\geq C_0h$ vertices, if the minimum semi-degree of $D$ is at least $\frac {n+h}{2}-1$, then $D$ contains a spanning $H$-subdivision. The lower bound on the minimum semi-degree is best possible. Furthermore, we show that there exist constants $C\gt 0$ and $\alpha , \beta \in (0, 1)$ such that for any integer partition $n=n_1+\cdots +n_m\geq Cm$ with $n_i\geq |V(H)|+3h$ for each $i$, and $\sum _{n_i\lt \alpha n}n_i\leq \beta n$, if a digraph of order $n\geq Cm$ has minimum semi-degree at least $\frac {n+m+h}{2}-1$, then it contains $m$ vertex-disjoint $H$-subdivisions whose orders are $n_1, \ldots , n_m$, respectively. The bound $\frac {n+m+h}{2}-1$ is also optimal. This work partly answers a conjecture of Lee [Combin. Probab. Comput. 34 (2025), 421–444] and generalises a recent result from the same paper.