The maximum size of an r-uniform hypergraph without a Berge cycle of length at least k has been determined for all k ≥ r + 3 by Füredi, Kostochka and Luo and for k < r (and k = r, asymptotically) by Kostochka and Luo. In this paper we settle the remaining cases: k = r + 1 and k = r + 2, proving a conjecture of Füredi, Kostochka and Luo.

An equitable colouring of a graph G is a vertex colouring where no two adjacent vertices are coloured the same and, additionally, the colour class sizes differ by at most 1. The equitable chromatic number χ=(G) is the minimum number of colours required for this. We study the equitable chromatic number of the dense random graph ${\mathcal{G}(n,m)}$ where $m = \left\lfloor {p\left( \matrix{n \cr 2 \cr}\right)} \right\rfloor $ and 0 < p < 0.86 is constant. It is a well-known question of Bollobás [3] whether for p = 1/2 there is a function f(n) → ∞ such that, for any sequence of intervals of length f(n), the normal chromatic number of ${\mathcal{G}(n,m)}$ lies outside the intervals with probability at least 1/2 if n is large enough. Bollobás proposes that this is likely to hold for f(n) = log n. We show that for the equitable chromatic number, the answer to the analogous question is negative. In fact, there is a subsequence ${({n_j})_j}_{ \in {\mathbb {N}}}$ of the integers where $\chi_=({\mathcal{G}(n_j,m_j)})$ is concentrated on exactly one explicitly known value. This constitutes surprisingly narrow concentration since in this range the equitable chromatic number, like the normal chromatic number, is rather large in absolute value, namely asymptotically equal to n/(2logbn) where b = 1/(1 − p).

We prove that n plane algebraic curves determine O(n(k+2)/(k+1)) points of kth order tangency. This generalizes an earlier result of Ellenberg, Solymosi and Zahl on the number of (first order) tangencies determined by n plane algebraic curves.

The inducibility of a graph H measures the maximum number of induced copies of H a large graph G can have. Generalizing this notion, we study how many induced subgraphs of fixed order k and size ℓ a large graph G on n vertices can have. Clearly, this number is $\left( {\matrix{n \cr k}}\right)$ for every n, k and $\ell \in \left\{ {0,\left( {\matrix{k \cr 2}} \right)}\right\}$. We conjecture that for every n, k and $0 \lt \ell \lt \left( {\matrix{k \cr 2}}\right)$ this number is at most $ (1/e + {o_k}(1)) {\left( {\matrix{n \cr k}} \right)}$. If true, this would be tight for ℓ ∈ {1, k − 1}.

In support of our ‘Edge-statistics Conjecture’, we prove that the corresponding density is bounded away from 1 by an absolute constant. Furthermore, for various ranges of the values of ℓ we establish stronger bounds. In particular, we prove that for ‘almost all’ pairs (k, ℓ) only a polynomially small fraction of the k-subsets of V(G) have exactly ℓ edges, and prove an upper bound of $ (1/2 + {o_k}(1)){\left( {\matrix{n \cr k}}\right)}$ for ℓ = 1.

Our proof methods involve probabilistic tools, such as anti-concentration results relying on fourth moment estimates and Brun’s sieve, as well as graph-theoretic and combinatorial arguments such as Zykov’s symmetrization, Sperner’s theorem and various counting techniques.

We prove that, with high probability, in every 2-edge-colouring of the random tournament on n vertices there is a monochromatic copy of every oriented tree of order $O(n{\rm{/}}\sqrt {{\rm{log}} \ n} )$. This generalizes a result of the first, third and fourth authors, who proved the same statement for paths, and is tight up to a constant factor.

We show that a nearly square independent and identically distributed random integral matrix is surjective over the integral lattice with very high probability. This answers a question by Koplewitz [6]. Our result extends to sparse matrices as well as to matrices of dependent entries.

For a rumour spreading protocol, the spread time is defined as the first time everyone learns the rumour. We compare the synchronous push&pull rumour spreading protocol with its asynchronous variant, and show that for any n-vertex graph and any starting vertex, the ratio between their expected spread times is bounded by $O({n^{1/3}}{\log ^{2/3}}n)$. This improves the $O(\sqrt n)$ upper bound of Giakkoupis, Nazari and Woelfel (2016). Our bound is tight up to a factor of O(log n), as illustrated by the string of diamonds graph. We also show that if, for a pair α, β of real numbers, there exist infinitely many graphs for which the two spread times are nα and nβ in expectation, then $0 \le \alpha \le 1$ and $\alpha \le \beta \le {1 \over 3} + {2 \over 3} \alpha $; and we show each such pair α, β is achievable.

We study random composite structures considered up to symmetry that are sampled according to weights on the inner and outer structures. This model may be viewed as an unlabelled version of Gibbs partitions and encompasses multisets of weighted combinatorial objects. We describe a general setting characterized by the formation of a giant component. The collection of small fragments is shown to converge in total variation toward a limit object following a Pólya–Boltzmann distribution.

The Brownian separable permuton is a random probability measure on the unit square, which was introduced by Bassino, Bouvel, Féray, Gerin and Pierrot (2016) as the scaling limit of the diagram of the uniform separable permutation as size grows to infinity. We show that, almost surely, the permuton is the pushforward of the Lebesgue measure on the graph of a random measure-preserving function associated to a Brownian excursion whose strict local minima are decorated with independent and identically distributed signs. As a consequence, its support is almost surely totally disconnected, has Hausdorff dimension one, and enjoys self-similarity properties inherited from those of the Brownian excursion. The density function of the averaged permuton is computed and a connection with the shuffling of the Brownian continuum random tree is explored.

Akbari and Alipour [1] conjectured that any Latin array of order n with at least n2/2 symbols contains a transversal. For large n, we confirm this conjecture, and moreover, we show that n399/200 symbols suffice.

The Friedgut–Kalai–Naor (FKN) theorem states that if ƒ is a Boolean function on the Boolean cube which is close to degree one, then ƒ is close to a dictator, a function depending on a single coordinate. The author has extended the theorem to the slice, the subset of the Boolean cube consisting of all vectors with fixed Hamming weight. We extend the theorem further, to the multislice, a multicoloured version of the slice.

As an application, we prove a stability version of the edge-isoperimetric inequality for settings of parameters in which the optimal set is a dictator.

The hedgehog Ht is a 3-uniform hypergraph on vertices $1, \ldots ,t + \left({\matrix{t \cr 2}}\right)$ such that, for any pair (i, j) with 1 ≤ i < j ≤ t, there exists a unique vertex k > t such that {i, j, k} is an edge. Conlon, Fox and Rödl proved that the two-colour Ramsey number of the hedgehog grows polynomially in the number of its vertices, while the four-colour Ramsey number grows exponentially in the square root of the number of vertices. They asked whether the two-colour Ramsey number of the hedgehog Ht is nearly linear in the number of its vertices. We answer this question affirmatively, proving that r(Ht) = O(t2 ln t).

For $$\tau \in {S_3}$$, let $$\mu _n^\tau $$ denote the uniformly random probability measure on the set of $$\tau $$-avoiding permutations in $${S_n}$$. Let $${\mathbb {N}^*} = {\mathbb {N}} \cup \{ \infty \} $$ with an appropriate metric and denote by $$S({\mathbb{N}},{\mathbb{N}^*})$$ the compact metric space consisting of functions $$\sigma {\rm{= }}\{ {\sigma _i}\} _{i = 1}^\infty {\rm{}}$$ from $$\mathbb {N}$$ to $${\mathbb {N}^ * }$$ which are injections when restricted to $${\sigma ^{ - 1}}(\mathbb {N})$$; that is, if $${\sigma _i}{\rm{= }}{\sigma _j}$$, $$i \ne j$$, then $${\sigma _i} = \infty $$. Extending permutations $$\sigma \in {S_n}$$ by defining $${\sigma _j} = j$$, for $$j \gt n$$, we have $${S_n} \subset S({\mathbb{N}},{{\mathbb{N}}^*})$$. For each $$\tau \in {S_3}$$, we study the limiting behaviour of the measures $$\{ \mu _n^\tau \} _{n = 1}^\infty $$ on $$S({\mathbb{N}},{\mathbb{N}^*})$$. We obtain partial results for the permutation $$\tau= 321$$ and complete results for the other five permutations $$\tau \in {S_3}$$.

We prove that, for q odd, a set of q + 2 points in the projective plane over the field with q elements has at least 2q − c odd secants, where c is a constant and an odd secant is a line incident with an odd number of points of the set.

Motivated by the work of Razborov about the minimal density of triangles in graphs we study the minimal density of the 5-cycle C5. We show that every graph of order n and size $ (1 - 1/k) \left( {\matrix{n \cr 2 }} \right) $, where k ≥ 3 is an integer, contains at least

$$({1 \over {10}} - {1 \over {2k}} + {1 \over {{k^2}}} - {1 \over {{k^3}}} + {2 \over {5{k^4}}}){n^5} + o({n^5})$$

The famous Erdős–Gallai theorem on the Turán number of paths states that every graph with n vertices and m edges contains a path with at least (2m)/n edges. In this note, we first establish a simple but novel extension of the Erdős–Gallai theorem by proving that every graph G contains a path with at least

$${{(s + 1){N_{s + 1}}(G)} \over {{N_s}(G)}} + s - 1$$

Given a graph G and a bijection f : E(G) → {1, 2,…,e(G)}, we say that a trail/path in G is f-increasing if the labels of consecutive edges of this trail/path form an increasing sequence. More than 40 years ago Chvátal and Komlós raised the question of providing worst-case estimates of the length of the longest increasing trail/path over all edge orderings of Kn. The case of a trail was resolved by Graham and Kleitman, who proved that the answer is n-1, and the case of a path is still wide open. Recently Lavrov and Loh proposed studying the average-case version of this problem, in which the edge ordering is chosen uniformly at random. They conjectured (and Martinsson later proved) that such an ordering with high probability (w.h.p.) contains an increasing Hamilton path.

In this paper we consider the random graph G = Gn,p with an edge ordering chosen uniformly at random. In this setting we determine w.h.p. the asymptotics of the number of edges in the longest increasing trail. In particular we prove an average-case version of the result of Graham and Kleitman, showing that the random edge ordering of Kn has w.h.p. an increasing trail of length (1-o(1))en, and that this is tight. We also obtain an asymptotically tight result for the length of the longest increasing path for random Erdős-Renyi graphs with p = o(1).

The induced removal lemma of Alon, Fischer, Krivelevich and Szegedy states that if an n-vertex graph G is ε-far from being induced H-free then G contains δH(ε) · nh induced copies of H. Improving upon the original proof, Conlon and Fox proved that 1/δH(ε)is at most a tower of height poly(1/ε), and asked if this bound can be further improved to a tower of height log(1/ε). In this paper we obtain such a bound for graphs G of density O(ε). We actually prove a more general result, which, as a special case, also gives a new proof of Fox’s bound for the (non-induced) removal lemma.

In an earlier paper, the present authors (2015) introduced the altermatic number of graphs and used Tucker’s lemma, an equivalent combinatorial version of the Borsuk–Ulam theorem, to prove that the altermatic number is a lower bound for chromatic number. A matching Kneser graph is a graph whose vertex set consists of all matchings of a specified size in a host graph and two vertices are adjacent if their corresponding matchings are edge-disjoint. Some well-known families of graphs such as Kneser graphs, Schrijver graphs and permutation graphs can be represented by matching Kneser graphs. In this paper, unifying and generalizing some earlier works by Lovász (1978) and Schrijver (1978), we determine the chromatic number of a large family of matching Kneser graphs by specifying their altermatic number. In particular, we determine the chromatic number of these matching Kneser graphs in terms of the generalized Turán number of matchings.