arXiv subjects

Andrew Treglown

Publications and source records attributed to Andrew Treglown.

At least 37 records · Page 2Linked to original sources

arXiv2021

On deficiency problems for graphs

Motivated by analogous questions in the setting of Steiner triple systems and Latin squares, Nenadov, Sudakov and Wagner [Completion and deficiency problems, Journal of Combinatorial Theory Series B, 2020] recently introduced the notion of graph deficiency. Given a global spanning property $\mathcal P$ and a graph $G$, the deficiency $\text{def}(G)$ of the graph $G$ with respect to the property $\mathcal P$ is the smallest non-negative integer $t$ such that the join $G*K_t$ has property $\mathcal P$. In particular, Nenadov, Sudakov and Wagner raised the question of determining how many edges an $n$-vertex graph $G$ needs to ensure $G*K_t$ contains a $K_r$-factor (for any fixed $r\geq 3$). In this paper we resolve their problem fully. We also give an analogous result which forces $G*K_t$ to contain any fixed bipartite $(n+t)$-vertex graph of bounded degree and small bandwidth.

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arXiv2021

Complete subgraphs in a multipartite graph

In 1975 Bollobás, Erd\H os, and Szemerédi asked the following question: given positive integers $n, t, r$ with $2\le t\le r-1$, what is the largest minimum degree $δ(G)$ among all $r$-partite graphs $G$ with parts of size $n$ and which do not contain a copy of $K_{t+1}$? The $r=t+1$ case has attracted a lot of attention and was fully resolved by Haxell and Szabó, and Szabó and Tardos in 2006. In this paper we investigate the $r>t+1$ case of the problem, which has remained dormant for over forty years. We resolve the problem exactly in the case when $r \equiv -1 \pmod{t}$, and up to an additive constant for many other cases, including when $r \geq (3t-1)(t-1)$. Our approach utilizes a connection to the related problem of determining the maximum of the minimum degrees among the family of balanced $r$-partite $rn$-vertex graphs of chromatic number at most $t$.

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arXiv2021

An asymmetric random Rado theorem for single equations: the $0$-statement

A famous result of Rado characterises those integer matrices $A$ which are partition regular, i.e. for which any finite colouring of the positive integers gives rise to a monochromatic solution to the equation $Ax=0$. Aigner-Horev and Person recently stated a conjecture on the probability threshold for the binomial random set $[n]_p$ having the asymmetric random Rado property: given partition regular matrices $A_1, \dots, A_r$ (for a fixed $r \geq 2$), however one $r$-colours $[n]_p$, there is always a colour $i \in [r]$ such that there is an $i$-coloured solution to $A_i x=0$. This generalises the symmetric case, which was resolved by Rödl and Ruciński, and Friedgut, Rödl and Schacht. Aigner-Horev and Person proved the $1$-statement of their asymmetric conjecture. In this paper, we resolve the $0$-statement in the case where the $A_i x=0$ correspond to single linear equations. Additionally we close a gap in the original proof of the 0-statement of the (symmetric) random Rado theorem.

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arXiv2021

A note on colour-bias Hamilton cycles in dense graphs

Balogh, Csaba, Jing and Pluhár recently determined the minimum degree threshold that ensures a $2$-coloured graph $G$ contains a Hamilton cycle of significant colour bias (i.e., a Hamilton cycle that contains significantly more than half of its edges in one colour). In this short note we extend this result, determining the corresponding threshold for $r$-colourings.

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arXiv2020

A discrepancy version of the Hajnal-Szemerédi theorem

A perfect $K_r$-tiling in a graph $G$ is a collection of vertex-disjoint copies of the clique $K_r$ in $G$ covering every vertex of $G$. The famous Hajnal--Szemerédi theorem determines the minimum degree threshold for forcing a perfect $K_r$-tiling in a graph $G$. The notion of discrepancy appears in many branches of mathematics. In the graph setting, one assigns the edges of a graph $G$ labels from $\{-1,1\}$, and one seeks substructures $F$ of $G$ that have `high' discrepancy (i.e. the sum of the labels of the edges in $F$ is far from $0$). In this paper we determine the minimum degree threshold for a graph to contain a perfect $K_r$-tiling of high discrepancy.

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arXiv2020

Tilings in randomly perturbed graphs: bridging the gap between Hajnal-Szemerédi and Johansson-Kahn-Vu

A perfect $K_r$-tiling in a graph $G$ is a collection of vertex-disjoint copies of $K_r$ that together cover all the vertices in $G$. In this paper we consider perfect $K_r$-tilings in the setting of randomly perturbed graphs; a model introduced by Bohman, Frieze and Martin where one starts with a dense graph and then adds $m$ random edges to it. Specifically, given any fixed $0< α<1-1/r$ we determine how many random edges one must add to an $n$-vertex graph $G$ of minimum degree $δ(G) \geq αn$ to ensure that, asymptotically almost surely, the resulting graph contains a perfect $K_r$-tiling. As one increases $α$ we demonstrate that the number of random edges required `jumps' at regular intervals, and within these intervals our result is best-possible. This work therefore closes the gap between the seminal work of Johansson, Kahn and Vu (which resolves the purely random case, i.e., $α=0$) and that of Hajnal and Szemerédi (which demonstrates that for $α\geq 1-1/r$ the initial graph already houses the desired perfect $K_r$-tiling).

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arXiv2020

Existence thresholds and Ramsey properties of random posets

Let $\mathcal P(n)$ denote the power set of $[n]$, ordered by inclusion, and let $\mathcal P (n,p)$ denote the random poset obtained from $\mathcal P(n)$ by retaining each element from $\mathcal P (n)$ independently at random with probability $p$ and discarding it otherwise. Given any fixed poset $F$ we determine the threshold for the property that $\mathcal P(n,p)$ contains $F$ as an induced subposet. We also asymptotically determine the number of copies of a fixed poset $F$ in $\mathcal P(n)$. Finally, we obtain a number of results on the Ramsey properties of the random poset $\mathcal P(n,p)$.

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arXiv2020

The bandwidth theorem for locally dense graphs

The Bandwidth theorem of Böttcher, Schacht and Taraz gives a condition on the minimum degree of an $n$-vertex graph $G$ that ensures $G$ contains every $r$-chromatic graph $H$ on $n$ vertices of bounded degree and of bandwidth $o(n)$, thereby proving a conjecture of Bollobás and Komlós. In this paper we prove a version of the Bandwidth theorem for locally dense graphs. Indeed, we prove that every locally dense $n$-vertex graph $G$ with $δ(G) > (1/2+o(1))n$ contains as a subgraph any given (spanning) $H$ with bounded maximum degree and sublinear bandwidth.

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arXiv2020

Transitive tournament tilings in oriented graphs with large minimum total degree

Let $\vec{T}_k$ be the transitive tournament on $k$ vertices. We show that every oriented graph on $n=4m$ vertices with minimum total degree $(11/12+o(1))n$ can be partitioned into vertex disjoint $\vec{T}_4$'s, and this bound is asymptotically tight. We also improve the best known bound on the minimum total degree for partitioning oriented graphs into vertex disjoint $\vec{T}_k$'s.

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arXiv2020

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 $t\ge 4$).

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arXiv2019

A degree sequence version of the Kühn-Osthus tiling theorem

A fundamental result of Kühn and Osthus [The minimum degree threshold for perfect graph packings, Combinatorica, 2009] determines up to an additive constant the minimum degree threshold that forces a graph to contain a perfect H-tiling. We prove a degree sequence version of this result which allows for a significant number of vertices to have lower degree.

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arXiv2019

A degree sequence Komlós theorem

An important result of Komlós [Tiling Turán theorems, Combinatorica, 2000] yields the asymptotically exact minimum degree threshold that ensures a graph $G$ contains an $H$-tiling covering an $x$th proportion of the vertices of $G$ (for any fixed $x \in (0,1)$ and graph $H$). We give a degree sequence strengthening of this result which allows for a large proportion of the vertices in the host graph $G$ to have degree substantially smaller than that required by Komlós' theorem. We also demonstrate that for certain graphs $H$, the degree sequence condition is essentially best possible in more than one sense.

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arXiv2018

Independent sets in hypergraphs and Ramsey properties of graphs and the integers

Many important problems in combinatorics and other related areas can be phrased in the language of independent sets in hypergraphs. Recently Balogh, Morris and Samotij, and independently Saxton and Thomason developed very general container theorems for independent sets in hypergraphs; both of which have seen numerous applications to a wide range of problems. In this paper we use the container method to give relatively short and elementary proofs of a number of results concerning Ramsey (and Turán) properties of (hyper)graphs and the integers. In particular: (i) We generalise the random Ramsey theorem of Rödl and Ruciński by providing a resilience analogue. Our result unifies and generalises several fundamental results in the area including the random version of Turán's theorem due to Conlon and Gowers and Schacht. (ii) The above result also resolves a general subcase of the asymmetric random Ramsey conjecture of Kohayakawa and Kreuter. (iii) All of the above results in fact hold for uniform hypergraphs. (iv) For a (hyper)graph $H$, we determine, up to an error term in the exponent, the number of $n$-vertex (hyper)graphs $G$ that have the Ramsey property with respect to $H$ (that is, whenever $G$ is $r$-coloured, there is a monochromatic copy of $H$ in $G$). (v) We strengthen the random Rado theorem of Friedgut, Rödl and Schacht by proving a resilience version of the result. (vi) For partition regular matrices $A$ we determine, up to an error term in the exponent, the number of subsets of $\{1,\dots,n\}$ for which there exists an $r$-colouring which contains no monochromatic solutions to $Ax=0$. Along the way a number of open problems are posed.

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arXiv2018

Sharp bound on the number of maximal sum-free subsets of integers

Cameron and Erdős asked whether the number of \emph{maximal} sum-free sets in $\{1, \dots , n\}$ is much smaller than the number of sum-free sets. In the same paper they gave a lower bound of $2^{\lfloor n/4 \rfloor }$ for the number of maximal sum-free sets. Here, we prove the following: For each $1\leq i \leq 4$, there is a constant $C_i$ such that, given any $n\equiv i \mod 4$, $\{1, \dots , n\}$ contains $(C_i+o(1)) 2^{n/4}$ maximal sum-free sets. Our proof makes use of container and removal lemmas of Green, a structural result of Deshouillers, Freiman, Sós and Temkin and a recent bound on the number of subsets of integers with small sumset by Green and Morris. We also discuss related results and open problems on the number of maximal sum-free subsets of abelian groups.

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arXiv2018

Tilings in randomly perturbed dense graphs

A perfect $H$-tiling in a graph $G$ is a collection of vertex-disjoint copies of a graph $H$ in $G$ that together cover all the vertices in $G$. In this paper we investigate perfect $H$-tilings in a random graph model introduced by Bohman, Frieze and Martin in which one starts with a dense graph and then adds $m$ random edges to it. Specifically, for any fixed graph $H$, we determine the number of random edges required to add to an arbitrary graph of linear minimum degree in order to ensure the resulting graph contains a perfect $H$-tiling with high probability. Our proof utilises Szemerédi's Regularity lemma as well as a special case of a result of Komlós concerning almost perfect $H$-tilings in dense graphs.

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arXiv2018

Applications of graph containers in the Boolean lattice

We apply the graph container method to prove a number of counting results for the Boolean lattice $\mathcal P(n)$. In particular, we: (i) Give a partial answer to a question of Sapozhenko estimating the number of $t$ error correcting codes in $\mathcal P(n)$, and we also give an upper bound on the number of transportation codes; (ii) Provide an alternative proof of Kleitman's theorem on the number of antichains in $\mathcal P(n)$ and give a two-coloured analogue; (iii) Give an asymptotic formula for the number of $(p,q)$-tilted Sperner families in $\mathcal P(n)$; (iv) Prove a random version of Katona's $t$-intersection theorem. In each case, to apply the container method, we first prove corresponding supersaturation results. We also give a construction which disproves two conjectures of Ilinca and Kahn on maximal independent sets and antichains in the Boolean lattice. A number of open questions are also given.

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arXiv2017

On solution-free sets of integers II

Given a linear equation $\mathcal{L}$, a set $A \subseteq [n]$ is $\mathcal{L}$-free if $A$ does not contain any `non-trivial' solutions to $\mathcal{L}$. We determine the precise size of the largest $\mathcal{L}$-free subset of $[n]$ for several general classes of linear equations $\mathcal{L}$ of the form $px+qy=rz$ for fixed $p,q,r \in \mathbb N$ where $p \geq q \geq r$. Further, for all such linear equations $\mathcal{L}$, we give an upper bound on the number of maximal $\mathcal{L}$-free subsets of $[n]$. In the case when $p=q\geq 2$ and $r=1$ this bound is exact up to an error term in the exponent. We make use of container and removal lemmas of Green to prove this result. Our results also extend to various linear equations with more than three variables.

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arXiv2017

An improved lower bound for Folkman's theorem

Folkman's Theorem asserts that for each $k \in \mathbb{N}$, there exists a natural number $n = F(k)$ such that whenever the elements of $[n]$ are two-coloured, there exists a set $A \subset [n]$ of size $k$ with the property that all the sums of the form $\sum_{x \in B} x$, where $B$ is a nonempty subset of $A$, are contained in $[n]$ and have the same colour. In 1989, Erdős and Spencer showed that $F(k) \ge 2^{ck^2/ \log k}$, where $c >0$ is an absolute constant; here, we improve this bound significantly by showing that $F(k) \ge 2^{2^{k-1}/k}$ for all $k\in \mathbb{N}$.

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