We prove that, for any finite set of minimal r-graph patterns, there is a finite family $\mathcal F$ of forbidden r-graphs such that the extremal Turán constructions for $\mathcal F$ are precisely the maximum r-graphs obtainable from mixing the given patterns in any way via blowups and recursion. This extends the result by the second author [30], where the above statement was established for a single pattern.

We present two applications of this result. First, we construct a finite family $\mathcal F$ of $3$-graphs such that there are exponentially many maximum $\mathcal F$-free $3$-graphs of each large order n and, moreover, the corresponding Turán problem is not finitely stable. Second, we show that there exists a finite family $\mathcal {F}$ of $3$-graphs whose feasible region function attains its maximum on a Cantor-type set of positive Hausdorff dimension.

We address a core partition regularity problem in Ramsey theory by proving that every finite coloring of the positive integers contains monochromatic Pythagorean pairs (i.e., $x,y\in {\mathbb N}$ such that $x^2\pm y^2=z^2$ for some $z\in {\mathbb N}$). We also show that partitions generated by level sets of multiplicative functions taking finitely many values always contain Pythagorean triples. Our proofs combine known Gowers uniformity properties of aperiodic multiplicative functions with a novel and rather flexible approach based on concentration estimates of multiplicative functions.

A left-variable word over an alphabet A is a word over $A \cup \{\star \}$ whose first letter is the distinguished symbol $\star $ standing for a placeholder. The ordered variable word theorem ($\mathsf {OVW}$), also known as Carlson–Simpson’s theorem, is a tree partition theorem, stating that for every finite alphabet A and every finite coloring of the words over A, there exists a word $c_0$ and an infinite sequence of left-variable words $w_1, w_2, \dots $ such that $\{ c_0 \cdot w_1[a_1] \cdot \dots \cdot w_k[a_k] : k \in \mathbb {N}, a_1, \dots , a_k \in A \}$ is monochromatic.

In this article, we prove that $\mathsf {OVW}$ is $\Pi ^0_4$-conservative over $\mathsf {RCA}_0 + \mathsf {B}\Sigma ^0_2$. This implies in particular that $\mathsf {OVW}$ does not imply $\mathsf {ACA}_0$ over $\mathsf {RCA}_0$. This is the first principle for which the only known separation from $\mathsf {ACA}_0$ involves non-standard models.

In [15], using methods from ergodic theory, a longstanding conjecture of Erdős (see [5, Page 305]) about sumsets in large subsets of the natural numbers was resolved. In this paper, we extend this result to several important classes of amenable groups, including all finitely generated virtually nilpotent groups and all abelian groups $(G,+)$ with the property that the subgroup $2G := \{g+g : g\in G\}$ has finite index. We prove that in any group G from the above classes, any $A\subset G$ with positive upper Banach density contains a shifted product set of the form $\{tb_ib_j\colon i<j\}$, for some infinite sequence $(b_n)_{n\in \mathbb {N}}$ and some $t\in G$. In fact, we show this result for all amenable groups that posses a property which we call square absolute continuity. Our results provide answers to several questions and conjectures posed in [13].

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.

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.

An extension of Szemerédi’s theorem is proved for sets of positive density in approximate lattices in general locally compact and second countable abelian groups. As a consequence, we establish a recent conjecture of Klick, Strungaru and Tcaciuc. Via a novel version of Furstenberg’s correspondence principle, which should be of independent interest, we show that our Szemerédi theorems can be deduced from a general transverse multiple recurrence theorem, which we establish using a recent work of Austin [Non-conventional ergodic averages for several commuting actions of an amenable group. J. Anal. Math. 130 (2016), 243–274].

We prove a new lower bound for the almost 20-year-old problem of determining the smallest possible size of an essential cover of the $n$-dimensional hypercube $\{\pm 1\}^n$, that is, the smallest possible size of a collection of hyperplanes that forms a minimal cover of $\{\pm 1\}^n$ and such that, furthermore, every variable appears with a non-zero coefficient in at least one of the hyperplane equations. We show that such an essential cover must consist of at least $10^{-2}\cdot n^{2/3}/(\log n)^{2/3}$ hyperplanes, improving previous lower bounds of Linial–Radhakrishnan, of Yehuda–Yehudayoff, and of Araujo–Balogh–Mattos.

A classical problem, due to Gerencsér and Gyárfás from 1967, asks how large a monochromatic connected component can we guarantee in any r-edge colouring of $K_n$? We consider how big a connected component we can guarantee in any r-edge colouring of $K_n$ if we allow ourselves to use up to s colours. This is actually an instance of a more general question of Bollobás from about 20 years ago which asks for a k-connected subgraph in the same setting. We complete the picture in terms of the approximate behaviour of the answer by determining it up to a logarithmic term, provided n is large enough. We obtain more precise results for certain regimes which solve a problem of Liu, Morris and Prince from 2007, as well as disprove a conjecture they pose in a strong form.

We also consider a generalisation in a similar direction of a question first considered by Erdős and Rényi in 1956, who considered given n and m, what is the smallest number of m-cliques which can cover all edges of $K_n$? This problem is essentially equivalent to the question of what is the minimum number of vertices that are certain to be incident to at least one edge of some colour in any r-edge colouring of $K_n$. We consider what happens if we allow ourselves to use up to s colours. We obtain a more complete understanding of the answer to this question for large n, in particular, determining it up to a constant factor for all $1\le s \le r$, as well as obtaining much more precise results for various ranges including the correct asymptotics for essentially the whole range.

We study density and partition properties of polynomial equations in prime variables. We consider equations of the form $a_1h(x_1) + \cdots + a_sh(x_s)=b$, where the ai and b are fixed coefficients and h is an arbitrary integer polynomial of degree d. We establish that the natural necessary conditions for this equation to have a monochromatic non-constant solution with respect to any finite colouring of the prime numbers are also sufficient when the equation has at least $(1+o(1))d^2$ variables. We similarly characterize when such equations admit solutions over any set of primes with positive relative upper density. In both cases, we obtain lower bounds for the number of monochromatic or dense solutions in primes that are of the correct order of magnitude. Our main new ingredient is a uniform lower bound on the cardinality of a prime polynomial Bohr set.

Inspired by Adler’s idea on VC minimal theories [1], we introduce VC-minimal complexity. We show that for any $N\in \mathbb {N}^{>0}$, there is $k_N>0$ such that for any finite bipartite graph $(X,Y;E)$ with VC-minimal complexity $< N$, there exist $X'\subseteq X$, $Y'\subseteq Y$ with $|X'|\geq k_N |X|$, $|Y'|\geq k_N |Y|$ such that $X'\times Y' \subseteq E$ or $X'\times Y'\cap E=\emptyset $.

We extend the Kechris–Pestov–Todorčević correspondence to weak Fraïssé categories and automorphism groups of generic objects. The new ingredient is the weak Ramsey property. We demonstrate the theory on several examples including monoid categories, the category of almost linear orders and categories of strong embeddings of trees.

Let $r_5(N)$ be the largest cardinality of a set in $\{1,\ldots,N\}$ which does not contain 5 elements in arithmetic progression. Then there exists a constant $c\in (0,1)$ such that

\[r_5(N)\ll \frac{N}{\exp\!((\!\log\log N)^{c})}.\]

$U^4$

We study covering numbers of subsets of the symmetric group $S_n$ that exhibit closure under conjugation, known as normal sets. We show that for any $\epsilon>0$, there exists $n_0$ such that if $n>n_0$ and A is a normal subset of the symmetric group $S_n$ of density $\ge e^{-n^{2/5 - \epsilon }}$, then $A^2 \supseteq A_n$. This improves upon a seminal result of Larsen and Shalev (Inventiones Math., 2008), with our $2/5$ in the double exponent replacing their $1/4$.

Our proof strategy combines two types of techniques. The first is ‘traditional’ techniques rooted in character bounds and asymptotics for the Witten zeta function, drawing from the foundational works of Liebeck–Shalev, Larsen–Shalev, and more recently, Larsen–Tiep. The second is a sharp hypercontractivity theorem in the symmetric group, which was recently obtained by Keevash and Lifshitz. This synthesis of algebraic and analytic methodologies not only allows us to attain our improved bounds but also provides new insights into the behavior of general independent sets in normal Cayley graphs over symmetric groups.

We show that every finite coloring of the rationals contains monochromatic sets of the form $\{x,y,xy,x+y\}$.

Given a family of graphs $\mathcal{F}$ and an integer $r$, we say that a graph is $r$-Ramsey for $\mathcal{F}$ if any $r$-colouring of its edges admits a monochromatic copy of a graph from $\mathcal{F}$. The threshold for the classic Ramsey property, where $\mathcal{F}$ consists of one graph, in the binomial random graph was located in the celebrated work of Rödl and Ruciński.

In this paper, we offer a twofold generalisation to the Rödl–Ruciński theorem. First, we show that the list-colouring version of the property has the same threshold. Second, we extend this result to finite families $\mathcal{F}$, where the threshold statements might also diverge. This also confirms further special cases of the Kohayakawa–Kreuter conjecture. Along the way, we supply a short(-ish), self-contained proof of the $0$-statement of the Rödl–Ruciński theorem.

Daisies are a special type of hypergraph introduced by Bollobás, Leader and Malvenuto. An $r$-daisy determined by a pair of disjoint sets $K$ and $M$ is the $(r+|K|)$-uniform hypergraph $\{K\cup P\,{:}\, P\in M^{(r)}\}$. Bollobás, Leader and Malvenuto initiated the study of Turán type density problems for daisies. This paper deals with Ramsey numbers of daisies, which are natural generalisations of classical Ramsey numbers. We discuss upper and lower bounds for the Ramsey number of $r$-daisies and also for special cases where the size of the kernel is bounded.

A $(k+r)$-uniform hypergraph $H$ on $(k+m)$ vertices is an $(r,m,k)$-daisy if there exists a partition of the vertices $V(H)=K\cup M$ with $|K|=k$, $|M|=m$ such that the set of edges of $H$ is all the $(k+r)$-tuples $K\cup P$, where $P$ is an $r$-tuple of $M$. We obtain an $(r-2)$-iterated exponential lower bound to the Ramsey number of an $(r,m,k)$-daisy for $2$-colours. This matches the order of magnitude of the best lower bounds for the Ramsey number of a complete $r$-graph.

We give a simple method to estimate the number of distinct copies of some classes of spanning subgraphs in hypergraphs with a high minimum degree. In particular, for each $k\geq 2$ and $1\leq \ell \leq k-1$, we show that every $k$-graph on $n$ vertices with minimum codegree at least

\begin{equation*} \left \{\begin {array}{l@{\quad}l} \left (\dfrac {1}{2}+o(1)\right )n & \text { if }(k-\ell )\mid k,\\[5pt] \left (\dfrac {1}{\lceil \frac {k}{k-\ell }\rceil (k-\ell )}+o(1)\right )n & \text { if }(k-\ell )\nmid k, \end {array} \right . \end{equation*}

$\exp\!(n\log n-\Theta (n))$

$\ell$

$(k-\ell )\mid n$

$(k-\ell )\mid k$

$(k-\ell )\nmid k$

$\ell \lt k/2$