Let us say that a graph $G$ is Ramsey for a tuple $(H_1,\ldots,H_r)$ of graphs if every r-colouring of the edges of G contains a monochromatic copy of $H_i$ in colour i, for some $i \in [\![{r}]\!]$. A famous conjecture of Kohayakawa and Kreuter, extending seminal work of Rödl and Ruciński, predicts the threshold at which the binomial random graph $G_{n,p}$ becomes Ramsey for $(H_1,\ldots,H_r)$ asymptotically almost surely.

In this paper, we resolve the Kohayakawa–Kreuter conjecture for almost all tuples of graphs. Moreover, we reduce its validity to the truth of a certain deterministic statement, which is a clear necessary condition for the conjecture to hold. All of our results actually hold in greater generality, when one replaces the graphs $H_1,\ldots,H_r$ by finite families $\mathcal{H}_1,\ldots,\mathcal{H}_r$. Additionally, we pose a natural (deterministic) graph-partitioning conjecture, which we believe to be of independent interest, and whose resolution would imply the Kohayakawa–Kreuter conjecture.

A classical result of Erdős, Lovász and Spencer from the late 1970s asserts that the dimension of the feasible region of densities of graphs with at most k vertices in large graphs is equal to the number of non-trivial connected graphs with at most k vertices. Indecomposable permutations play the role of connected graphs in the realm of permutations, and Glebov et al. showed that pattern densities of indecomposable permutations are independent, i.e., the dimension of the feasible region of densities of permutation patterns of size at most k is at least the number of non-trivial indecomposable permutations of size at most k. However, this lower bound is not tight already for $k=3$. We prove that the dimension of the feasible region of densities of permutation patterns of size at most k is equal to the number of non-trivial Lyndon permutations of size at most k. The proof exploits an interplay between algebra and combinatorics inherent to the study of Lyndon words.

We provide two methods to characterise the connectedness of all d-dimensional generalised Sierpiński sponges whose corresponding iterated function systems (IFSs) are allowed to have rotational and reflectional components. Our approach is to reduce it to an intersection problem between the coordinates of graph-directed attractors. More precisely, let $(K_1,\ldots,K_n)$ be a Cantor-type graph-directed attractor in ${\mathbb {R}}^d$. By creating an auxiliary graph, we provide an effective criterion for whether $K_i\cap K_j$ is empty for every pair of $1\leq i,j\leq n$. Moreover, the emptiness can be checked by examining only a finite number of geometric approximations of the attractor. The approach is also applicable to more general graph-directed systems.

Given a self-morphism $\phi$ on a projective variety defined over a number field k, we prove two results which bound the largest iterate of $\phi$ whose evaluation at P is quasi-integral with respect to a divisor D, uniformly across P defined over a field of bounded degree over k. The first result applies when the pullback of D by some iterate of $\phi$ breaks up into enough irreducible components which are numerical multiples of each other. The proof uses Le’s algebraic-point version of a result of Ji–Yan–Yu, which is based on Schmidt subspace theorem. The second result applies more generally but relies on a deep conjecture by Vojta for algebraic points. The second result is an extension of a recent result of Matsuzawa, based on the theory of asymptotic multiplicity. Both results are generalisations of Hsia–Silverman, which treated the case of morphisms on ${\mathbb{P}}^1$.

Given a fixed k-uniform hypergraph F, the F-removal lemma states that every hypergraph with few copies of F can be made F-free by the removal of few edges. Unfortunately, for general F, the constants involved are given by incredibly fast-growing Ackermann-type functions. It is thus natural to ask for which F one can prove removal lemmas with polynomial bounds. One trivial case where such bounds can be obtained is when F is k-partite. Alon proved that when $k=2$ (i.e. when dealing with graphs), only bipartite graphs have a polynomial removal lemma. Kohayakawa, Nagle and Rödl conjectured in 2002 that Alon’s result can be extended to all $k\gt2$, namely, that the only $k$-graphs $F$ for which the hypergraph removal lemma has polynomial bounds are the trivial cases when F is k-partite. In this paper we prove this conjecture.

Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface of genus $g \geq 1$, and let $\mathrm{LMod}_{p}(X)$ be the liftable mapping class group associated with a finite-sheeted branched cover $p:S \to X$, where X is a hyperbolic surface. For $k \geq 2$, let $p_k: S_{k(g-1)+1} \to S_g$ be the standard k-sheeted regular cyclic cover. In this paper, we show that $\{\mathrm{LMod}_{p_k}(S_g)\}_{k \geq 2}$ forms an infinite family of self-normalising subgroups in $\mathrm{Mod}(S_g)$, which are also maximal when k is prime. Furthermore, we derive explicit finite generating sets for $\mathrm{LMod}_{p_k}(S_g)$ for $g \geq 3$ and $k \geq 2$, and $\mathrm{LMod}_{p_2}(S_2)$. For $g \geq 2$, as an application of our main result, we also derive a generating set for $\mathrm{LMod}_{p_2}(S_g) \cap C_{\mathrm{Mod}(S_g)}(\iota)$, where $C_{\mathrm{Mod}(S_g)}(\iota)$ is the centraliser of the hyperelliptic involution $\iota \in \mathrm{Mod}(S_g)$. Let $\mathcal{L}$ be the infinite ladder surface, and let $q_g : \mathcal{L} \to S_g$ be the standard infinite-sheeted cover induced by $\langle h^{g-1} \rangle$ where h is the standard handle shift on $\mathcal{L}$. As a final application, we derive a finite generating set for $\mathrm{LMod}_{q_g}(S_g)$ for $g \geq 3$.

We describe several exotic fusion systems related to the sporadic simple groups at odd primes. More generally, we classify saturated fusion systems supported on Sylow 3-subgroups of the Conway group $\textrm{Co}_1$ and the Thompson group $\textrm{F}_3$, and a Sylow 5-subgroup of the Monster M, as well as a particular maximal subgroup of the latter two p-groups. This work is supported by computations in MAGMA.

The q-colour Ramsey number of a k-uniform hypergraph H is the minimum integer N such that any q-colouring of the complete k-uniform hypergraph on N vertices contains a monochromatic copy of H. The study of these numbers is one of the central topics in Combinatorics. In 1973, Erdős and Graham asked to maximise the Ramsey number of a graph as a function of the number of its edges. Motivated by this problem, we study the analogous question for hypergaphs. For fixed $k \ge 3$ and $q \ge 2$ we prove that the largest possible q-colour Ramsey number of a k-uniform hypergraph with m edges is at most $\mathrm{tw}_k(O(\sqrt{m})),$ where tw denotes the tower function. We also present a construction showing that this bound is tight for $q \ge 4$. This resolves a problem by Conlon, Fox and Sudakov. They previously proved the upper bound for $k \geq 4$ and the lower bound for $k=3$. Although in the graph case the tightness follows simply by considering a clique of appropriate size, for higher uniformities the construction is rather involved and is obtained by using paths in expander graphs.

We construct skew corner-free subsets of $[n]^2$ of size $n^2\exp(\!-O(\sqrt{\log n}))$, thereby improving on recent bounds of the form $\Omega(n^{5/4})$ obtained by Pohoata and Zakharov. We also prove that any such set has size at most $O(n^2(\log n)^{-c})$ for some absolute constant $c \gt 0$. This improves on the previously best known upper bound $O(n^2(\log\log n)^{-c})$, coming from Shkredov’s work on the corners theorem.

In this paper, we establish homological Berglund–Hübsch mirror symmetry for curve singularities where the A–model incorporates equivariance, otherwise known as homological Berglund–Hübsch–Henningson mirror symmetry, including for certain deformations of categories. More precisely, we prove a conjecture of Futaki and Ueda which posits that the equivariance in the A–model can be incorporated by pulling back the superpotential to the total space of the corresponding crepant resolution. Along the way, we show that the B–model category of matrix factorisations has a tilting object whose length is the dimension of the state space of the Fan–Jarvis–Ruan–Witten (FJRW) A–model, a result which might be of independent interest for its implications in the Landau–Ginzburg analogue of Dubrovin’s conjecture.

Given a surface $\Sigma$ equipped with a set P of marked points, we consider the triangulations of $\Sigma$ with vertex set P. The flip-graph of $\Sigma$ is the graph whose vertices are these triangulations, and whose edges correspond to flipping arcs in these triangulations. The flip-graph of a surface appears in the study of moduli spaces and mapping class groups. We consider the number of geodesics in the flip-graph of $\Sigma$ between two triangulations as a function of their distance. We show that this number grows exponentially provided the surface has enough topology, and that in the remaining cases the growth is polynomial.

In an earlier work, we defined a “generalised Temperley–Lieb algebra” $TL_{r, 1, n}$ corresponding to the imprimitive reflection group G(r, 1, n) as a quotient of the cyclotomic Hecke algebra. In this work we introduce the generalised Temperley–Lieb algebra $TL_{r, p, n}$ which corresponds to the complex reflection group G(r, p, n). Our definition identifies $TL_{r, p, n}$ as the fixed-point subalgebra of $TL_{r, 1, n}$ under a certain automorphism $\sigma$. We prove the cellularity of $TL_{r, p, n}$ by proving that $\sigma$ induces a special shift automorphism with respect to the cellular structure of $TL_{r, 1, n}$. We also give a description of the cell modules of $TL_{r, p, n}$ and their decomposition numbers, and finally we point to how our algebras might be categorified and could lead to a diagrammatic theory.

We prove new statistical results about the distribution of the cokernel of a random integral matrix with a concentrated residue. Given a prime p and a positive integer n, consider a random $n \times n$ matrix $X_n$ over the ring $\mathbb{Z}_p$ of p-adic integers whose entries are independent. Previously, Wood showed that as long as each entry of $X_n$ is not too concentrated on a single residue modulo p, regardless of its distribution, the distribution of the cokernel $\mathrm{cok}(X_n)$ of $X_n$, up to isomorphism, weakly converges to the Cohen–Lenstra distribution, as $n \rightarrow \infty$. Here on the contrary, we consider the case when $X_n$ has a concentrated residue $A_n$ so that $X_n = A_n + pB_n$. When $B_n$ is a Haar-random $n \times n$ matrix over $\mathbb{Z}_p$, we explicitly compute the distribution of $\mathrm{cok}(P(X_n))$ for every fixed n and a non-constant monic polynomial $P(t) \in \mathbb{Z}_p[t]$. We deduce our result from an interesting equidistribution result for matrices over $\mathbb{Z}_p[t]/(P(t))$, which we prove by establishing a version of the Weierstrass preparation theorem for the noncommutative ring $\mathrm{M}_n(\mathbb{Z}_p)$ of $n \times n$ matrices over $\mathbb{Z}_p$. We also show through cases the subtlety of the “universality” behavior when $B_n$ is not Haar-random.

We present a construction of left braces of right nilpotency class at most two based on suitable actions of an abelian group on itself with an invariance condition. This construction allows us to recover the construction of a free right nilpotent one-generated left brace of class two.

The aim of this paper is to establish the correspondence between the twisted localised Pestov identity on the unit tangent bundle of a Riemannian manifold and the Weitzenböck identity for twisted symmetric tensors on the manifold.

The space of monic squarefree complex polynomials has a stratification according to the multiplicities of the critical points. We introduce a method to study these strata by way of the infinite-area translation surface associated to the logarithmic derivative $df/f$ of the polynomial. We determine the monodromy of these strata in the braid group, thus describing which braidings of the roots are possible if the orders of the critical points are required to stay fixed. Mirroring the story for holomorphic differentials on higher-genus surfaces, we find the answer is governed by the framing of the punctured disk induced by the horizontal foliation on the translation surface.

Let $b \geqslant 3$ be an integer and C(b, D) be the set of real numbers in [0,1] whose base b expansion only consists of digits in a set $D {\subseteq} \{0,...,b-1\}$. We study how close can numbers in C(b, D) be approximated by rational numbers with denominators being powers of some integer t and obtain a zero-full law for its Hausdorff measure in several circumstances. When b and t are multiplicatively dependent, our results correct an error of Levesley, Salp and Velani (Math. Ann. 338 (2007), 97–118) and generalise their theorem. When b and t are multiplicatively independent but have the same prime divisors, we obtain a partial result on the Hausdorff measure and bounds for the Hausdorff dimension, which are close to the multiplicatively dependent case. Based on these results, several conjectures are proposed.

Let $B^{H}$ be a d-dimensional fractional Brownian motion with Hurst index $H\in(0,1)$, $f\,:\,[0,1]\longrightarrow\mathbb{R}^{d}$ a Borel function, and $E\subset[0,1]$, $F\subset\mathbb{R}^{d}$ are given Borel sets. The focus of this paper is on hitting probabilities of the non-centered Gaussian process $B^{H}+f$. It aims to highlight how each component f, E and F is involved in determining the upper and lower bounds of $\mathbb{P}\{(B^H+f)(E)\cap F\neq \emptyset \}$. When F is a singleton and f is a general measurable drift, some new estimates are obtained for the last probability by means of suitable Hausdorff measure and capacity of the graph $Gr_E(f)$. As application we deal with the issue of polarity of points for $(B^H+f)\vert_E$ (the restriction of $B^H+f$ to the subset $E\subset (0,\infty)$).