For given graphs G1,…, Gk, the size-Ramsey number $\hat R({G_1}, \ldots ,{G_k})$ is the smallest integer m for which there exists a graph H on m edges such that in every k-edge colouring of H with colours 1,…,k, H contains a monochromatic copy of Gi of colour i for some 1 ≤ i ≤ k. We denote $\hat R({G_1}, \ldots ,{G_k})$ by ${\hat R_k}(G)$ when G1 = ⋯ = Gk = G.

Haxell, Kohayakawa and Łuczak showed that the size-Ramsey number of a cycle Cn is linear in n, ${\hat R_k}({C_n}) \le {c_k}n$ for some constant ck. Their proof, however, is based on Szemerédi’s regularity lemma so no specific constant ck is known.

In this paper, we give various upper bounds for the size-Ramsey numbers of cycles. We provide an alternative proof of ${\hat R_k}({C_n}) \le {c_k}n$, avoiding use of the regularity lemma, where ck is exponential and doubly exponential in k, when n is even and odd, respectively. In particular, we show that for sufficiently large n we have ${\hat R_2}({C_n}) \le {10^5} \times cn$, where c = 6.5 if n is even and c = 1989 otherwise.

May the triforce be the 3-uniform hypergraph on six vertices with edges {123′, 12′3, 1′23}. We show that the minimum triforce density in a 3-uniform hypergraph of edge density δ is δ4–o(1) but not O(δ4).

Let M(δ) be the maximum number such that the following holds: for every ∊ > 0 and $G = {\mathbb{F}}_2^n$ with n sufficiently large, if A ⊆ G × G with A ≥ δ|G|2, then there exists a nonzero “popular difference” d ∈ G such that the number of “corners” (x, y), (x + d, y), (x, y + d) ∈ A is at least (M(δ)–∊)|G|2. As a corollary via a recent result of Mandache, we conclude that M(δ) = δ4–o(1) and M(δ) = ω(δ4).

On the other hand, for 0 < δ < 1/2 and sufficiently large N, there exists A ⊆ [N]3 with |A| ≥ δN3 such that for every d ≠ 0, the number of corners (x, y, z), (x + d, y, z), (x, y + d, z), (x, y, z + d) ∈ A is at most δc log(1/δ)N3. A similar bound holds in higher dimensions, or for any configuration with at least 5 points or affine dimension at least 3.

Let $n$ be a positive integer and $a$ an integer prime to $n$. Multiplication by $a$ induces a permutation over $\mathbb{Z}/n\mathbb{Z}=\{\overline{0},\overline{1},\ldots ,\overline{n-1}\}$. Lerch’s theorem gives the sign of this permutation. We explore some applications of Lerch’s result to permutation problems involving quadratic residues modulo $p$ and confirm some conjectures posed by Sun [‘Quadratic residues and related permutations and identities’, Preprint, 2018, arXiv:1809.07766]. We also study permutations involving arbitrary $k$th power residues modulo $p$ and primitive roots modulo a power of $p$.

Given two k-graphs (k-uniform hypergraphs) F and H, a perfect F-tiling (or F-factor) in H is a set of vertex-disjoint copies of F that together cover the vertex set of H. For all complete k-partite k-graphs K, Mycroft proved a minimum codegree condition that guarantees a K-factor in an n-vertex k-graph, which is tight up to an error term o(n). In this paper we improve the error term in Mycroft’s result to a sublinear term that relates to the Turán number of K when the differences of the sizes of the vertex classes of K are co-prime. Furthermore, we find a construction which shows that our improved codegree condition is asymptotically tight in infinitely many cases, thus disproving a conjecture of Mycroft. Finally, we determine exact minimum codegree conditions for tiling K(k)(1, … , 1, 2) and tiling loose cycles, thus generalizing the results of Czygrinow, DeBiasio and Nagle, and of Czygrinow, respectively.

Consider any fixed graph whose edges have been randomly and independently oriented, and write {S ⇝} to indicate that there is an oriented path going from a vertex s ∊ S to vertex i. Narayanan (2016) proved that for any set S and any two vertices i and j, {S ⇝ i} and {S ⇝ j} are positively correlated. His proof relies on the Ahlswede–Daykin inequality, a rather advanced tool of probabilistic combinatorics.

In this short note I give an elementary proof of the following, stronger result: writing V for the vertex set of the graph, for any source set S, the events {S ⇝ i}, i ∊ V, are positively associated, meaning that the expectation of the product of increasing functionals of the family {S ⇝ i} for i ∊ V is greater than the product of their expectations.

The conjecture of Brown, Erdős and Sós from 1973 states that, for any k ≥ 3, if a 3-uniform hypergraph H with n vertices does not contain a set of k +3 vertices spanning at least k edges then it has o(n2) edges. The case k = 3 of this conjecture is the celebrated (6, 3)-theorem of Ruzsa and Szemerédi which implies Roth’s theorem on 3-term arithmetic progressions in dense sets of integers. Solymosi observed that, in order to prove the conjecture, one can assume that H consists of triples (a, b, ab) of some finite quasigroup Γ. Since this problem remains open for all k ≥ 4, he further proposed to study triple systems coming from finite groups. In this case he proved that the conjecture holds also for k = 4. Here we completely resolve the Brown–Erdős–Sós conjecture for all finite groups and values of k. Moreover, we prove that the hypergraphs coming from groups contain sets of size $\Theta (\sqrt k )$ which span k edges. This is best possible and goes far beyond the conjecture.

We take an elementary and systematic approach to the problem of extending the Tutte polynomial to the setting of embedded graphs. Four notions of embedded graphs arise naturally when considering deletion and contraction operations on graphs on surfaces. We give a description of each class in terms of coloured ribbon graphs. We then identify a universal deletion-contraction invariant (i.e., a ‘Tutte polynomial’) for each class. We relate these to graph polynomials in the literature, including the Bollobás–Riordan, Krushkal and Las Vergnas polynomials, and give state-sum formulations, duality relations, deleton-contraction relations, and quasi-tree expansions for each of them.

Let r ⩾ 2 be a fixed constant and let $ {\cal H} $ be an r-uniform, D-regular hypergraph on N vertices. Assume further that D → ∞ as N → ∞ and that degrees of pairs of vertices in $ {\cal H} $ are at most L where L = D/( log N)ω(1). We consider the random greedy algorithm for forming a matching in $ {\cal H} $. We choose a matching at random by iteratively choosing edges uniformly at random to be in the matching and deleting all edges that share at least one vertex with a chosen edge before moving on to the next choice. This process terminates when there are no edges remaining in the graph. We show that with high probability the proportion of vertices of $ {\cal H} $ that are not saturated by the final matching is at most (L/D)(1/(2(r−1)))+o(1). This point is a natural barrier in the analysis of the random greedy hypergraph matching process.

A family of sets is said to be intersecting if any two sets in the family have non-empty intersection. In 1973, Erdős raised the problem of determining the maximum possible size of a union of r different intersecting families of k-element subsets of an n-element set, for each triple of integers (n, k, r). We make progress on this problem, proving that for any fixed integer r ⩾ 2 and for any $$k \le ({1 \over 2} - o(1))n$$, if X is an n-element set, and $${\cal F} = {\cal F}_1 \cup {\cal F}_2 \cup \cdots \cup {\cal F}_r $$, where each $$ {\cal F}_i $$ is an intersecting family of k-element subsets of X, then $$|{\cal F}| \le \left( {\matrix{n \cr k \cr } } \right) - \left( {\matrix{{n - r} \cr k \cr } } \right)$$, with equality only if $${\cal F} = \{ S \subset X:|S| = k,\;S \cap R \ne \emptyset \} $$ for some R ⊂ X with |R| = r. This is best possible up to the size of the o(1) term, and improves a 1987 result of Frankl and Füredi, who obtained the same conclusion under the stronger hypothesis $$k < (3 - \sqrt 5 )n/2$$, in the case r = 2. Our proof utilizes an isoperimetric, influence-based method recently developed by Keller and the authors.

A family of sets is intersecting if no two of its members are disjoint, and has the Erdős–Ko–Rado property (or is EKR) if each of its largest intersecting subfamilies has non-empty intersection.

Denote by ${{\cal H}_k}(n,p)$ the random family in which each k-subset of {1, …, n} is present with probability p, independent of other choices. A question first studied by Balogh, Bohman and Mubayi asks:

\begin{equation} {\rm{For what }}p = p(n,k){\rm{is}}{{\cal H}_k}(n,p){\rm{likely to be EKR}}? \end{equation}

Here, for fixed c < 1/4, and $k \lt \sqrt {cn\log n} $ we give a precise answer to this question, characterizing those sequences p = p(n, k) for which

\begin{equation} {\mathbb{P}}({{\cal H}_k}(n,p){\rm{is EKR}}{\kern 1pt} ) \to 1{\rm{as }}n \to \infty . \end{equation}

We completely classify Cartan subalgebras of dimension drop algebras with coprime parameters. More generally, we classify Cartan subalgebras of arbitrary stabilised dimension drop algebras that are non-degenerate in the sense that the dimensions of their fibres in the endpoints are maximal. Conjugacy classes by an automorphism are parametrised by certain congruence classes of matrices over the natural numbers with prescribed row and column sums. In particular, each dimension drop algebra admits only finitely many non-degenerate Cartan subalgebras up to conjugacy. As a consequence of this parametrisation, we can provide examples of subhomogeneous $\text{C}^{\ast }$-algebras with exactly $n$ Cartan subalgebras up to conjugacy. Moreover, we show that in many dimension drop algebras two Cartan subalgebras are conjugate if and only if their spectra are homeomorphic.

We conjecture that bounded generalised polynomial functions cannot be generated by finite automata, except for the trivial case when they are ultimately periodic.

Using methods from ergodic theory, we are able to partially resolve this conjecture, proving that any hypothetical counterexample is periodic away from a very sparse and structured set. In particular, we show that for a polynomial $p(n)$ with at least one irrational coefficient (except for the constant one) and integer $m\geqslant 2$, the sequence $\lfloor p(n)\rfloor \hspace{0.2em}{\rm mod}\hspace{0.2em}m$ is never automatic.

We also prove that the conjecture is equivalent to the claim that the set of powers of an integer $k\geqslant 2$ is not given by a generalised polynomial.

We analyse the asymptotic extremal growth rate of the Betti numbers of clique complexes of graphs on n vertices not containing a fixed forbidden induced subgraph H.

In particular, we prove a theorem of the alternative: for any H the growth rate achieves exactly one of five possible exponentials, that is, independent of the field of coefficients, the nth root of the maximal total Betti number over n-vertex graphs with no induced copy of H has a limit, as n tends to infinity, and, ranging over all H, exactly five different limits are attained.

For the interesting case where H is the 4-cycle, the above limit is 1, and we prove a superpolynomial upper bound.

Let m(k) denote the maximum number of edges in a non-extendable, intersecting k-graph. Erdős and Lovász proved that m(k) ≤ kk. For k ≥ 625 we prove m(k) < kk・e−k1/4/6.

Let $F$ be an algebraically closed field of characteristic $0$ and let $\operatorname{sp}(2l,F)$ be the rank $l$ symplectic algebra of all $2l\times 2l$ matrices $x=\big(\!\begin{smallmatrix}A & B\\ C & -A^{t}\end{smallmatrix}\!\big)$ over $F$, where $A^{t}$ is the transpose of $A$ and $B,C$ are symmetric matrices of order $l$. The commuting graph $\unicode[STIX]{x1D6E4}(\operatorname{sp}(2l,F))$ of $\operatorname{sp}(2l,F)$ is a graph whose vertex set consists of all nonzero elements in $\operatorname{sp}(2l,F)$ and two distinct vertices $x$ and $y$ are adjacent if and only if $xy=yx$. We prove that the diameter of $\unicode[STIX]{x1D6E4}(\operatorname{sp}(2l,F))$ is $4$ when $l>2$.

In 1987, Alavi, Malde, Schwenk and Erdős conjectured that the independence polynomial of any tree is unimodal. Although it attracts many researchers' attention, it is still open. Motivated by this conjecture, in this paper, we prove that rooted products of some graphs preserve real rootedness of independence polynomials. As application, we not only give a unified proof for some known results, but also we can apply them to generate infinite kinds of trees whose independence polynomials have only real zeros. Thus their independence polynomials are unimodal.

For positive integers $t_{1},\ldots ,t_{k}$, let $\tilde{p}(n,t_{1},t_{2},\ldots ,t_{k})$ (respectively $p(n,t_{1},t_{2},\ldots ,t_{k})$) be the number of partitions of $n$ such that, if $m$ is the smallest part, then each of $m+t_{1},m+t_{1}+t_{2},\ldots ,m+t_{1}+t_{2}+\cdots +t_{k-1}$ appears as a part and the largest part is at most (respectively equal to) $m+t_{1}+t_{2}+\cdots +t_{k}$. Andrews et al. [‘Partitions with fixed differences between largest and smallest parts’, Proc. Amer. Math. Soc.143 (2015), 4283–4289] found an explicit formula for the generating function of $p(n,t_{1},t_{2},\ldots ,t_{k})$. We establish a $q$-series identity from which the formulae for the generating functions of $\tilde{p}(n,t_{1},t_{2},\ldots ,t_{k})$ and $p(n,t_{1},t_{2},\ldots ,t_{k})$ can be obtained.

The category of Cohen–Macaulay modules of an algebra $B_{k,n}$ is used in Jensen et al. (A categorification of Grassmannian cluster algebras, Proc. Lond. Math. Soc. (3) 113(2) (2016), 185–212) to give an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of $k$-planes in $n$-space. In this paper, we find canonical Auslander–Reiten sequences and study the Auslander–Reiten translation periodicity for this category. Furthermore, we give an explicit construction of Cohen–Macaulay modules of arbitrary rank. We then use our results to establish a correspondence between rigid indecomposable modules of rank 2 and real roots of degree 2 for the associated Kac–Moody algebra in the tame cases.

We consider a random graph model that was recently proposed as a model for complex networks by Krioukov et al. (2010). In this model, nodes are chosen randomly inside a disk in the hyperbolic plane and two nodes are connected if they are at most a certain hyperbolic distance from each other. It has previously been shown that this model has various properties associated with complex networks, including a power-law degree distribution and a strictly positive clustering coefficient. The model is specified using three parameters: the number of nodes N, which we think of as going to infinity, and $\alpha, \nu > 0$, which we think of as constant. Roughly speaking, $\alpha$ controls the power-law exponent of the degree sequence and $\nu$ the average degree. Earlier work of Kiwi and Mitsche (2015) has shown that, when $\alpha \lt 1$ (which corresponds to the exponent of the power law degree sequence being $\lt 3$), the diameter of the largest component is asymptotically almost surely (a.a.s.) at most polylogarithmic in N. Friedrich and Krohmer (2015) showed it was a.a.s. $\Omega(\log N)$ and improved the exponent of the polynomial in $\log N$ in the upper bound. Here we show the maximum diameter over all components is a.a.s. $O(\log N),$ thus giving a bound that is tight up to a multiplicative constant.

Letting ℳ denote the space of finite measures on ℕ, and μλ ∊ ℳ denote the Poisson distribution with parameter λ, the function W : [0, 1]2 → ℳ given by W(x, y) = μc log x log y is called the PAG graphon with density c. It is known that this is the limit, in the multigraph homomorphism sense, of the dense preferential attachment graph (PAG) model with edge density c. This graphon can then in turn be used to generate the so-called W-random graphs in a natural way, and similar constructions also work in the slightly more general context of the so-called PAGκ models. The aim of this paper is to compare these dense PAGκ models with the W-random graph models obtained from the corresponding graphons. Motivated by the multigraph limit theory, we investigate the expected jumble-norm distance of the two models in terms of the number of vertices n. We present a coupling for which the expectation can be bounded from above by O(log3/2n · n−1/2), and provide a universal lower bound that is coupling-independent, but without the logarithmic term.