We present a stand-alone simple proof of a probabilistic interpretation of the Gaussian binomial coefficients by conditioning a random walk to hit a given lattice point at a given time.

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.

Given a finite group $G$ , the generating graph $\unicode[STIX]{x1D6E4}(G)$ of $G$ has as vertices the (nontrivial) elements of $G$ and two vertices are adjacent if and only if they are distinct and generate $G$ as group elements. In this paper we investigate properties about the degrees of the vertices of $\unicode[STIX]{x1D6E4}(G)$ when $G$ is an alternating group or a symmetric group of degree $n$ . In particular, we determine the vertices of $\unicode[STIX]{x1D6E4}(G)$ having even degree and show that $\unicode[STIX]{x1D6E4}(G)$ is Eulerian if and only if $n\geqslant 3$ and $n$ and $n-1$ are not equal to a prime number congruent to 3 modulo 4.

The classical Kővári–Sós–Turán theorem states that if G is an n-vertex graph with no copy of K s,t as a subgraph, then the number of edges in G is at most O(n 2−1/s ). We prove that if one forbids K s,t as an induced subgraph, and also forbids any fixed graph H as a (not necessarily induced) subgraph, the same asymptotic upper bound still holds, with different constant factors. This introduces a non-trivial angle from which to generalize Turán theory to induced forbidden subgraphs, which this paper explores. Along the way, we derive a non-trivial upper bound on the number of cliques of fixed order in a K r -free graph with no induced copy of K s,t . This result is an induced analogue of a recent theorem of Alon and Shikhelman and is of independent interest.

In network modelling of complex systems one is often required to sample random realizations of networks that obey a given set of constraints, usually in the form of graph measures. A much studied class of problems targets uniform sampling of simple graphs with given degree sequence or also with given degree correlations expressed in the form of a Joint Degree Matrix. One approach is to use Markov chains based on edge switches (swaps) that preserve the constraints, are irreducible (ergodic) and fast mixing. In 1999, Kannan, Tetali and Vempala (KTV) proposed a simple swap Markov chain for sampling graphs with given degree sequence, and conjectured that it mixes rapidly (in polynomial time) for arbitrary degree sequences. Although the conjecture is still open, it has been proved for special degree sequences, in particular for those of undirected and directed regular simple graphs, half-regular bipartite graphs, and graphs with certain bounded maximum degrees. Here we prove the fast mixing KTV conjecture for novel, exponentially large classes of irregular degree sequences. Our method is based on a canonical decomposition of degree sequences into split graph degree sequences, a structural theorem for the space of graph realizations and on a factorization theorem for Markov chains. After introducing bipartite ‘splitted’ degree sequences, we also generalize the canonical split graph decomposition for bipartite and directed graphs.

In this paper, it is proved that the complement of the zero-divisor graph of a partially ordered set is weakly perfect if it has finite clique number, completely answering the question raised by Joshi and Khiste [‘Complement of the zero divisor graph of a lattice’, Bull. Aust. Math. Soc. 89 (2014), 177–190]. As a consequence, the intersection graph of an intersection-closed family of nonempty subsets of a set is weakly perfect if it has finite clique number. These results are applied to annihilating-ideal graphs and intersection graphs of submodules.

An r-uniform hypergraph is called an r-graph. A hypergraph is linear if every two edges intersect in at most one vertex. Given a linear r-graph H and a positive integer n, the linear Turán number exL(n,H) is the maximum number of edges in a linear r-graph G that does not contain H as a subgraph. For each ℓ ≥ 3, let Crℓ denote the r-uniform linear cycle of length ℓ, which is an r-graph with edges e1, . . ., eℓ such that, for all i ∈ [ℓ−1], |ei ∩ ei+1|=1, |eℓ ∩ e1|=1 and ei ∩ ej = ∅ for all other pairs {i,j}, i ≠ j. For all r ≥ 3 and ℓ ≥ 3, we show that there exists a positive constant c = cr,ℓ, depending only r and ℓ, such that exL(n,Crℓ) ≤ cn1+1/⌊ℓ/2⌋. This answers a question of Kostochka, Mubayi and Verstraëte [30]. For even ℓ, our result extends the result of Bondy and Simonovits [7] on the Turán numbers of even cycles to linear hypergraphs.

Using our results on linear Turán numbers, we also obtain bounds on the cycle-complete hypergraph Ramsey numbers. We show that there are positive constants a = am,r and b = bm,r, depending only on m and r, such that

\begin{equation}R(C^r_{2m}, K^r_t)\leq a \Bigl(\frac{t}{\ln t}\Bigr)^{{m}/{(m-1)}}\quad\text{and}\quadR(C^r_{2m+1}, K^r_t)\leq b t^{{m}/{(m-1)}}.\end{equation}

It is proved that every non-trivial Latin square has an upper embedding in a non-orientable surface and every Latin square of odd order has an upper embedding in an orientable surface. In the latter case, detailed results about the possible automorphisms and their actions are also obtained.

Inspired by the recent work of M. Nakasuji, O. Phuksuwan and Y. Yamasaki, we combine interpolated multiple zeta values and Schur multiple zeta values into one object, which we call interpolated Schur multiple zeta values. Our main result will be a Jacobi–Trudi formula for a certain class of these new objects. This generalizes an analogous result for Schur multiple zeta values and implies algebraic relations between interpolated multiple zeta values.

We provide lower bounds for $p$ -adic valuations of multisums of factorial ratios which satisfy an Apéry-like recurrence relation: these include Apéry, Domb and Franel numbers, the numbers of abelian squares over a finite alphabet, and constant terms of powers of certain Laurent polynomials. In particular, we prove Beukers’ conjectures on the $p$ -adic valuation of Apéry numbers. Furthermore, we give an effective criterion for a sequence of factorial ratios to satisfy the $p$ -Lucas property for almost all primes $p$ .

We present an abstract framework for the axiomatic study of diagram algebras. Algebras that fit this framework possess analogues of both the Murphy and seminormal bases of the Hecke algebras of the symmetric groups. We show that the transition matrix between these bases is dominance unitriangular. We construct analogues of the skew Specht modules in this setting. This allows us to propose a natural tableaux theoretic framework in which to study the infamous Kronecker problem.

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 is part of the ongoing effort to study high-dimensional permutations. We prove the analogue to the Erdős–Szekeres theorem: For every k ≥ 1, every order-nk-dimensional permutation contains a monotone subsequence of length Ωk ( $\sqrt{n}$ ), and this is tight. On the other hand, and unlike the classical case, the longest monotone subsequence in a random k-dimensional permutation of order n is asymptotically almost surely Θk (n k/(k+1)).

Assume that the edges of the complete graph Kn are given independent uniform [0, 1] weights. We consider the expected minimum total weight μk of k ⩽ 2 edge-disjoint spanning trees. When k is large we show that μk ≈ k 2. Most of the paper is concerned with the case k = 2. We show that m 2 tends to an explicitly defined constant and that μ 2 ≈ 4.1704288. . . .

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

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

Let $G$ be a simple connected graph with $n$ vertices and $m$ edges and $d_{1}\geq d_{2}\geq \cdots \geq d_{n}>0$ its sequence of vertex degrees. If $\unicode[STIX]{x1D707}_{1}\geq \unicode[STIX]{x1D707}_{2}\geq \cdots \geq \unicode[STIX]{x1D707}_{n-1}>\unicode[STIX]{x1D707}_{n}=0$ are the Laplacian eigenvalues of $G$ , then the Kirchhoff index of $G$ is $\mathit{Kf}(G)=n\sum _{i=1}^{n-1}\unicode[STIX]{x1D707}_{i}^{-1}$ . We prove some new lower bounds for $\mathit{Kf}(G)$ in terms of some of the parameters $\unicode[STIX]{x1D6E5}=d_{1}$ , $\unicode[STIX]{x1D6E5}_{2}=d_{2}$ , $\unicode[STIX]{x1D6E5}_{3}=d_{3}$ , $\unicode[STIX]{x1D6FF}=d_{n}$ , $\unicode[STIX]{x1D6FF}_{2}=d_{n-1}$ and the topological index $\mathit{NK}=\prod _{i=1}^{n}d_{i}$ .

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.

Bollobás and Scott (Random Struct. Alg.21 (2002) 414–430) asked for conditions that guarantee a bisection of a graph with m edges in which each class has at most (1/4+o(1))m edges. We demonstrate that cycles of length 4 play an important role for this question. Let G be a graph with m edges, minimum degree δ, and containing no cycle of length 4. We show that if (i) G is 2-connected, or (ii) δ ⩾ 3, or (iii) δ ⩾ 2 and the girth of G is at least 5, then G admits a bisection in which each class has at most (1/4+o(1))m edges. We show that each of these conditions are best possible. On the other hand, a construction by Alon, Bollobás, Krivelevich and Sudakov shows that for infinitely many m there exists a graph with m edges and girth at least 5 for which any bisection has at least (1/4−o(1))m edges in one of the two classes.

The classification of flag-transitive generalized quadrangles is a long-standing open problem at the interface of finite geometry and permutation group theory. Given that all known flag-transitive generalized quadrangles are also point-primitive (up to point–line duality), it is likewise natural to seek a classification of the point-primitive examples. Working toward this aim, we are led to investigate generalized quadrangles that admit a collineation group $G$ preserving a Cartesian product decomposition of the set of points. It is shown that, under a generic assumption on $G$, the number of factors of such a Cartesian product can be at most four. This result is then used to treat various types of primitive and quasiprimitive point actions. In particular, it is shown that $G$ cannot have holomorph compound O’Nan–Scott type. Our arguments also pose purely group-theoretic questions about conjugacy classes in nonabelian finite simple groups and fixities of primitive permutation groups.

We consider community detection in degree-corrected stochastic block models. We propose a spectral clustering algorithm based on a suitably normalized adjacency matrix. We show that this algorithm consistently recovers the block membership of all but a vanishing fraction of nodes, in the regime where the lowest degree is of order log(n) or higher. Recovery succeeds even for very heterogeneous degree distributions. The algorithm does not rely on parameters as input. In particular, it does not need to know the number of communities.