On the complexity of finding and counting solution-free sets of integers

Given a linear equation $\mathcal{L}$, a set $A$ of integers is $\mathcal{L}$-free if $A$ does not contain any `non-trivial' solutions to $\mathcal{L}$. This notion incorporates many central topics in combinatorial number theory such as sum-free and progression-free sets. In this paper we initiate the study of (parameterised) complexity questions involving $\mathcal{L}$-free sets of integers. The main questions we consider involve deciding whether a finite set of integers $A$ has an $\mathcal{L}$-free subset of a given size, and counting all such $\mathcal{L}$-free subsets. We also raise a number of open problems.Comment: 27 page

Exact minimum degree thresholds for perfect matchings in uniform hypergraphs II

Given positive integers k\geq 3 and r where k/2 \leq r \leq k-1, we give a minimum r-degree condition that ensures a perfect matching in a k-uniform hypergraph. This condition is best possible and improves on work of Pikhurko who gave an asymptotically exact result. Our approach makes use of the absorbing method, and builds on work in 'Exact minimum degree thresholds for perfect matchings in uniform hypergraphs', where we proved the result for k divisible by 4.Comment: 20 pages, 1 figur

Ramsey properties of randomly perturbed graphs: cliques and cycles

Given graphs $H_1,H_2$, a graph $G$ is $(H_1,H_2)$-Ramsey if for every colouring of the edges of $G$ with red and blue, there is a red copy of $H_1$ or a blue copy of $H_2$. In this paper we investigate Ramsey questions in the setting of randomly perturbed graphs: this is a random graph model introduced by Bohman, Frieze and Martin in which one starts with a dense graph and then adds a given number of random edges to it. The study of Ramsey properties of randomly perturbed graphs was initiated by Krivelevich, Sudakov and Tetali in 2006; they determined how many random edges must be added to a dense graph to ensure the resulting graph is with high probability $(K_3,K_t)$-Ramsey (for $t\ge 3$). They also raised the question of generalising this result to pairs of graphs other than $(K_3,K_t)$. We make significant progress on this question, giving a precise solution in the case when $H_1=K_s$ and $H_2=K_t$ where $s,t \ge 5$. Although we again show that one requires polynomially fewer edges than in the purely random graph, our result shows that the problem in this case is quite different to the $(K_3,K_t)$-Ramsey question. Moreover, we give bounds for the corresponding $(K_4,K_t)$-Ramsey question; together with a construction of Powierski this resolves the $(K_4,K_4)$-Ramsey problem. We also give a precise solution to the analogous question in the case when both $H_1=C_s$ and $H_2=C_t$ are cycles. Additionally we consider the corresponding multicolour problem. Our final result gives another generalisation of the Krivelevich, Sudakov and Tetali result. Specifically, we determine how many random edges must be added to a dense graph to ensure the resulting graph is with high probability $(C_s,K_t)$-Ramsey (for odd $s\ge 5$ and $t\ge 4$).Comment: 24 pages + 12-page appendix; v2: cited independent work of Emil Powierski, stated results for cliques in graphs of low positive density separately (Theorem 1.6) for clarity; v3: author accepted manuscript, to appear in CP

Matchings in 3-uniform hypergraphs

We determine the minimum vertex degree that ensures a perfect matching in a 3-uniform hypergraph. More precisely, suppose that H is a sufficiently large 3-uniform hypergraph whose order n is divisible by 3. If the minimum vertex degree of H is greater than \binom{n-1}{2}-\binom{2n/3}{2}, then H contains a perfect matching. This bound is tight and answers a question of Han, Person and Schacht. More generally, we show that H contains a matching of size d\le n/3 if its minimum vertex degree is greater than \binom{n-1}{2}-\binom{n-d}{2}, which is also best possible. This extends a result of Bollobas, Daykin and Erdos.Comment: 18 pages, 1 figure. To appear in JCT

An Ore-type theorem for perfect packings in graphs

We say that a graph G has a perfect H-packing (also called an H-factor) if there exists a set of disjoint copies of H in G which together cover all the vertices of G. Given a graph H, we determine, asymptotically, the Ore-type degree condition which ensures that a graph G has a perfect H-packing. More precisely, let \delta_{\rm Ore} (H,n) be the smallest number k such that every graph G whose order n is divisible by |H| and with d(x)+d(y)\geq k for all non-adjacent x \not = y \in V(G) contains a perfect H-packing. We determine \lim_{n\to \infty} \delta_{\rm Ore} (H,n)/n.Comment: 23 pages, 1 figure. Extra examples and a sketch proof of Theorem 4 added. To appear in the SIAM Journal on Discrete Mathematic