We prove that the number of multigraphs with vertex set {1, . . ., n} such that every four vertices span at most nine edges is an2+o(n2) where a is transcendental (assuming Schanuel's conjecture from number theory). This is an easy consequence of the solution to a related problem about maximizing the product of the edge multiplicities in certain multigraphs, and appears to be the first explicit (somewhat natural) question in extremal graph theory whose solution is transcendental. These results may shed light on a question of Razborov, who asked whether there are conjectures or theorems in extremal combinatorics which cannot be proved by a certain class of finite methods that include Cauchy–Schwarz arguments.

Our proof involves a novel application of Zykov symmetrization applied to multigraphs, a rather technical progressive induction, and a straightforward use of hypergraph containers.

A set of points in d-dimensional Euclidean space is almost equidistant if, among any three points of the set, some two are at distance 1. We show that an almost-equidistant set in ℝd has cardinality O(d4/3).

The goal of property testing is to quickly distinguish between objects which satisfy a property and objects that are ε-far from satisfying the property. There are now several general results in this area which show that natural properties of combinatorial objects can be tested with ‘constant’ query complexity, depending only on ε and the property, and not on the size of the object being tested. The upper bound on the query complexity coming from the proof techniques is often enormous and impractical. It remains a major open problem if better bounds hold.

Maybe surprisingly, for testing with respect to the rectangular distance, we prove there is a universal (not depending on the property), polynomial in 1/ε query complexity bound for two-sided testing hereditary properties of sufficiently large permutations. We further give a nearly linear bound with respect to a closely related metric which also depends on the smallest forbidden subpermutation for the property. Finally, we show that several different permutation metrics of interest are related to the rectangular distance, yielding similar results for testing with respect to these metrics.

For fixed integers p and q, let f(n,p,q) denote the minimum number of colours needed to colour all of the edges of the complete graph Kn such that no clique of p vertices spans fewer than q distinct colours. Any edge-colouring with this property is known as a (p,q)-colouring. We construct an explicit (5,5)-colouring that shows that f(n,5,5) ≤ n1/3 + o(1) as n → ∞. This improves upon the best known probabilistic upper bound of O(n1/2) given by Erdős and Gyárfás, and comes close to matching the best known lower bound Ω(n1/3).

We study the minimum degree necessary to guarantee the existence of perfect and almost-perfect triangle-tilings in an n-vertex graph G with sublinear independence number. In this setting, we show that if δ(G) ≥ n/3 + o(n), then G has a triangle-tiling covering all but at most four vertices. Also, for every r ≥ 5, we asymptotically determine the minimum degree threshold for a perfect triangle-tiling under the additional assumptions that G is Kr-free and n is divisible by 3.

A sunflower is a collection of distinct sets such that the intersection of any two of them is the same as the common intersection C of all of them, and |C| is smaller than each of the sets. A longstanding conjecture due to Erdős and Szemerédi (solved recently in [7, 9]; see also [22]) was that the maximum size of a family of subsets of [n] that contains no sunflower of fixed size k > 2 is exponentially smaller than 2n as n → ∞. We consider the problems of determining the maximum sum and product of k families of subsets of [n] that contain no sunflower of size k with one set from each family. For the sum, we prove that the maximum is

$$(k-1)2^n+1+\sum_{s=0}^{k-2}\binom{n}{s}$$

$$\biggl(\ffrac{1}{8}+o(1)\biggr)2^{3n}.$$

It follows from known results that every regular tripartite hypergraph of positive degree, with n vertices in each class, has matching number at least n/2. This bound is best possible, and the extremal configuration is unique. Here we prove a stability version of this statement, establishing that every regular tripartite hypergraph with matching number at most (1 + ϵ)n/2 is close in structure to the extremal configuration, where ‘closeness’ is measured by an explicit function of ϵ.

Keller and Kindler recently established a quantitative version of the famous Benjamini–Kalai–Schramm theorem on the noise sensitivity of Boolean functions. Their result was extended to the continuous Gaussian setting by Keller, Mossel and Sen by means of a Central Limit Theorem argument. In this work we present a unified approach to these results, in both discrete and continuous settings. The proof relies on semigroup decompositions together with a suitable cut-off argument, allowing for the efficient use of the classical hypercontractivity tool behind these results. It extends to further models of interest such as families of log-concave measures and Cayley and Schreier graphs. In particular we obtain a quantitative version of the Benjamini–Kalai–Schramm theorem for the slices of the Boolean cube.

In 2006 Brown asked the following question in the spirit of Ramsey theory: given a non-periodic infinite word $x=x_{1}x_{2}x_{3}\ldots$ with values in a set $\mathbb{A}$, does there exist a finite colouring $\unicode[STIX]{x1D711}:\mathbb{A}^{+}\rightarrow C$ relative to which $x$ does not admit a $\unicode[STIX]{x1D711}$-monochromatic factorization, i.e. a factorization of the form $x=u_{1}u_{2}u_{3}\ldots$ with $\unicode[STIX]{x1D711}(u_{i})=\unicode[STIX]{x1D711}(u_{\!j})$ for all $i,j\geqslant 1$? Various partial results in support of an affirmative answer to this question have appeared in the literature in recent years. In particular it is known that the question admits an affirmative answer for all non-uniformly recurrent words and for various classes of uniformly recurrent words including Sturmian words and fixed points of strongly recognizable primitive substitutions. In this paper we give a complete and optimal affirmative answer to this question by showing that if $x=x_{1}x_{2}x_{3}\ldots$ is an infinite non-periodic word with values in a set $\mathbb{A}$, then there exists a $2$-colouring $\unicode[STIX]{x1D711}:\mathbb{A}^{+}\rightarrow \{0,1\}$ such that for any factorization $x=u_{1}u_{2}u_{3}\ldots$ we have $\unicode[STIX]{x1D711}(u_{i})\neq \unicode[STIX]{x1D711}(u_{\!j})$ for some $i\neq j$.

An oriented k-uniform hypergraph (a family of ordered k-sets) has the ordering property (or Property O) if, for every linear order of the vertex set, there is some edge oriented consistently with the linear order. We find bounds on the minimum number of edges in a hypergraph with Property O.

We use the probabilistic method to obtain versions of the colourful Carathéodory theorem and Tverberg's theorem with tolerance.

In particular, we give bounds for the smallest integer N = N(t,d,r) such that for any N points in ℝd, there is a partition of them into r parts for which the following condition holds: after removing any t points from the set, the convex hulls of what is left in each part intersect.

We prove a bound N = rt + O($\sqrt{t}$) for fixed r,d which is polynomial in each parameters. Our bounds extend to colourful versions of Tverberg's theorem, as well as Reay-type variations of this theorem.

A celebrated result of Rödl and Ruciński states that for every graph $F$, which is not a forest of stars and paths of length 3, and fixed number of colours $r\geqslant 2$ there exist positive constants $c,C$ such that for $p\leqslant cn^{-1/m_{2}(F)}$ the probability that every colouring of the edges of the random graph $G(n,p)$ contains a monochromatic copy of $F$ is $o(1)$ (the ‘0-statement’), while for $p\geqslant Cn^{-1/m_{2}(F)}$ it is $1-o(1)$ (the ‘1-statement’). Here $m_{2}(F)$ denotes the 2-density of $F$. On the other hand, the case where $F$ is a forest of stars has a coarse threshold which is determined by the appearance of a certain small subgraph in $G(n,p)$. Recently, the natural extension of the 1-statement of this theorem to $k$-uniform hypergraphs was proved by Conlon and Gowers and, independently, by Friedgut, Rödl and Schacht. In particular, they showed an upper bound of order $n^{-1/m_{k}(F)}$ for the 1-statement, where $m_{k}(F)$ denotes the $k$-density of $F$. Similarly as in the graph case, it is known that the threshold for star-like hypergraphs is given by the appearance of small subgraphs. In this paper we show that another type of threshold exists if $k\geqslant 4$: there are $k$-uniform hypergraphs for which the threshold is determined by the asymmetric Ramsey problem in which a different hypergraph has to be avoided in each colour class. Along the way we obtain a general bound on the 1-statement for asymmetric Ramsey properties in random hypergraphs. This extends the work of Kohayakawa and Kreuter, and of Kohayakawa, Schacht and Spöhel who showed a similar result in the graph case. We prove the corresponding 0-statement for hypergraphs satisfying certain balancedness conditions.

A family of subsets of {1,. . .,n} is called intersecting if any two of its sets intersect. A classical result in extremal combinatorics due to Erdős, Ko and Rado determines the maximum size of an intersecting family of k-subsets of {1,. . .,n}. In this paper we study the following problem: How many intersecting families of k-subsets of {1,. . .,n} are there? Improving a result of Balogh, Das, Delcourt, Liu and Sharifzadeh, we determine this quantity asymptotically for n ≥ 2k+2+2$\sqrt{k\log k}$ and k → ∞. Moreover, under the same assumptions we also determine asymptotically the number of non-trivial intersecting families, that is, intersecting families for which the intersection of all sets is empty. We obtain analogous results for pairs of cross-intersecting families.

This paper deals with a combinatorial problem concerning colourings of uniform hypergraphs with large girth. We prove that if H is an n-uniform non-r-colourable simple hypergraph then its maximum edge degree Δ(H) satisfies the inequality

$$ \Delta(H)\geqslant c\cdot r^{n-1}\ffrac{n(\ln\ln n)^2}{\ln n} $$

As an application of our probabilistic technique we establish a lower bound for the classical van der Waerden number W(n, r), the minimum natural N such that in an arbitrary colouring of the set of integers {1,. . .,N} with r colours there exists a monochromatic arithmetic progression of length n. We prove that

$$ W(n,r)\geqslant c\cdot r^{n-1}\ffrac{(\ln\ln n)^2}{\ln n}. $$

We prove that Boolean functions on $S_{n}$, whose Fourier transform is highly concentrated on irreducible representations indexed by partitions of $n$ whose largest part has size at least $n-t$, are close to being unions of cosets of stabilizers of $t$-tuples. We also obtain an edge-isoperimetric inequality for the transposition graph on $S_{n}$ which is asymptotically sharp for subsets of $S_{n}$ of size $n!/\text{poly}(n)$, using eigenvalue techniques. We then combine these two results to obtain a sharp edge-isoperimetric inequality for subsets of $S_{n}$ of size $(n-t)!$, where $n$ is large compared to $t$, confirming a conjecture of Ben Efraim in these cases.

A random binary search tree grown from the uniformly random permutation of [n] is studied. We analyze the exact and asymptotic counts of vertices by rank, the distance from the set of leaves. The asymptotic fraction ck of vertices of a fixed rank k ≥ 0 is shown to decay exponentially with k. We prove that the ranks of the uniformly random, fixed size sample of vertices are asymptotically independent, each having the distribution {ck}. Notoriously hard to compute, the exact fractions ck have been determined for k ≤ 3 only. We present a shortcut enabling us to compute c4 and c5 as well; both are ratios of enormous integers, the denominator of c5 being 274 digits long. Prompted by the data, we prove that, in sharp contrast, the largest prime divisor of the denominator of ck is at most 2k+1 + 1. We conjecture that, in fact, the prime divisors of every denominator for k > 1 form a single interval, from 2 to the largest prime not exceeding 2k+1 + 1.

In this paper we compute the absorbing time Tn of an n-dimensional discrete-time Markov chain comprising n components, each with an absorbing state and evolving in mutual exclusion. We show that the random absorbing time Tn is well approximated by a deterministic time tn that is the first time when a fluid approximation of the chain approaches the absorbing state at a distance 1 / n. We provide an asymptotic expansion of tn that uses the spectral decomposition of the kernel of the chain as well as the asymptotic distribution of Tn, relying on extreme values theory. We show the applicability of this approach with three different problems: the coupon collector, the erasure channel lifetime, and the coupling times of random walks in high-dimensional spaces.

Let hom(G) denote the size of the largest clique or independent set of a graph G. In 2007, Bukh and Sudakov proved that every n-vertex graph G with hom(G) = O(logn) contains an induced subgraph with Ω(n1/2) distinct degrees, and raised the question of deciding whether an analogous result holds for every n-vertex graph G with hom(G) = O(nϵ), where ϵ > 0 is a fixed constant. Here, we answer their question in the affirmative and show that every graph G on n vertices contains an induced subgraph with Ω((n/hom(G))1/2) distinct degrees. We also prove a stronger result for graphs with large cliques or independent sets and show, for any fixed k ∈ ℕ, that if an n-vertex graph G contains no induced subgraph with k distinct degrees, then hom(G)⩾n/(k − 1) − o(n); this bound is essentially best possible.

A collection of $k$ sets is said to form a $k$-sunflower, or $\unicode[STIX]{x1D6E5}$-system, if the intersection of any two sets from the collection is the same, and we call a family of sets ${\mathcal{F}}$sunflower-free if it contains no $3$-sunflowers. Following the recent breakthrough of Ellenberg and Gijswijt (‘On large subsets of $\mathbb{F}_{q}^{n}$ with no three-term arithmetic progression’, Ann. of Math. (2) 185 (2017), 339–343); (‘Progression-free sets in $\mathbb{Z}_{4}^{n}$ are exponentially small’, Ann. of Math. (2) 185 (2017), 331–337) we apply the polynomial method directly to Erdős–Szemerédi sunflower problem (Erdős and Szemerédi, ‘Combinatorial properties of systems of sets’, J. Combin. Theory Ser. A 24 (1978), 308–313) and prove that any sunflower-free family ${\mathcal{F}}$ of subsets of $\{1,2,\ldots ,n\}$ has size at most

$$\begin{eqnarray}|{\mathcal{F}}|\leqslant 3n\mathop{\sum }_{k\leqslant n/3}\binom{n}{k}\leqslant \left(\frac{3}{2^{2/3}}\right)^{n(1+o(1))}.\end{eqnarray}$$

$A\subset (\mathbb{Z}/D\mathbb{Z})^{n}=\{1,2,\ldots ,D\}^{n}$

$D>2$

$x,y,z\in A$

$i$

$x_{i},y_{i},z_{i}$

$\unicode[STIX]{x1D712}:\mathbb{Z}/D\mathbb{Z}\rightarrow \mathbb{C}$

$A\subset (\mathbb{Z}/D\mathbb{Z})^{n}$

$$\begin{eqnarray}|A|\leqslant c_{D}^{n}\end{eqnarray}$$

$c_{D}=\frac{3}{2^{2/3}}(D-1)^{2/3}$

$c_{D}\leqslant C$

$C$

$D$

Let G be a finite D-quasirandom group and A ⊂ Gk a δ-dense subset. Then the density of the set of side lengths g of corners

$$ \{(a_{1},\dotsc,a_{k}),(ga_{1},a_{2},\dotsc,a_{k}),\dotsc,(ga_{1},\dotsc,ga_{k})\} \subset A $$