arXiv subjects

David Conlon

Publications and source records attributed to David Conlon.

At least 73 records · Page 4Linked to original sources

arXiv2020

Ramsey games near the critical threshold

A well-known result of Rödl and Ruciński states that for any graph $H$ there exists a constant $C$ such that if $p \geq C n^{- 1/m_2(H)}$, then the random graph $G_{n,p}$ is a.a.s. $H$-Ramsey, that is, any $2$-colouring of its edges contains a monochromatic copy of $H$. Aside from a few simple exceptions, the corresponding $0$-statement also holds, that is, there exists $c>0$ such that whenever $p\leq cn^{-1/m_2(H)}$ the random graph $G_{n,p}$ is a.a.s. not $H$-Ramsey. We show that near this threshold, even when $G_{n,p}$ is not $H$-Ramsey, it is often extremely close to being $H$-Ramsey. More precisely, we prove that for any constant $c > 0$ and any strictly $2$-balanced graph $H$, if $p \geq c n^{-1/m_2(H)}$, then the random graph $G_{n,p}$ a.a.s. has the property that every $2$-edge-colouring without monochromatic copies of $H$ cannot be extended to an $H$-free colouring after $ω(1)$ extra random edges are added. This generalises a result by Friedgut, Kohayakawa, Rödl, Ruciński and Tetali, who in 2002 proved the same statement for triangles, and addresses a question raised by those authors. We also extend a result of theirs on the three-colour case and show that these theorems need not hold when $H$ is not strictly $2$-balanced.

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arXiv2020

Short proofs of some extremal results III

We prove a selection of results from different areas of extremal combinatorics, including complete or partial solutions to a number of open problems. These results, coming mainly from extremal graph theory and Ramsey theory, have been collected together because in each case the relevant proofs are reasonably short.

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arXiv2020

Hypergraph expanders of all uniformities from Cayley graphs

Hypergraph expanders are hypergraphs with surprising, non-intuitive expansion properties. In a recent paper, the first author gave a simple construction, which can be randomized, of $3$-uniform hypergraph expanders with polylogarithmic degree. We generalize this construction, giving a simple construction of $r$-uniform hypergraph expanders for all $r \geq 3$.

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arXiv2020

More on the extremal number of subdivisions

Given a graph $H$, the extremal number $\mathrm{ex}(n,H)$ is the largest number of edges in an $H$-free graph on $n$ vertices. We make progress on a number of conjectures about the extremal number of bipartite graphs. First, writing $K'_{s,t}$ for the subdivision of the bipartite graph $K_{s,t}$, we show that $\mathrm{ex}(n, K'_{s,t}) = O(n^{3/2 - \frac{1}{2s}})$. This proves a conjecture of Kang, Kim and Liu and is tight up to the implied constant for $t$ sufficiently large in terms of $s$. Second, for any integers $s, k \geq 1$, we show that $\mathrm{ex}(n, L) = Θ(n^{1 + \frac{s}{sk+1}})$ for a particular graph $L$ depending on $s$ and $k$, answering another question of Kang, Kim and Liu. This result touches upon an old conjecture of Erdős and Simonovits, which asserts that every rational number $r \in (1,2)$ is realisable in the sense that $\mathrm{ex}(n,H) = Θ(n^r)$ for some appropriate graph $H$, giving infinitely many new realisable exponents and implying that $1 + 1/k$ is a limit point of realisable exponents for all $k \geq 1$. Writing $H^k$ for the $k$-subdivision of a graph $H$, this result also implies that for any bipartite graph $H$ and any $k$, there exists $δ> 0$ such that $\mathrm{ex}(n,H^{k-1}) = O(n^{1 + 1/k - δ})$, partially resolving a question of Conlon and Lee. Third, extending a recent result of Conlon and Lee, we show that any bipartite graph $H$ with maximum degree $r$ on one side which does not contain $C_4$ as a subgraph satisfies $\mathrm{ex}(n, H) = o(n^{2 - 1/r})$.

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arXiv2019

The Ramsey number of books

We show that in every two-colouring of the edges of the complete graph $K_N$ there is a monochromatic $K_k$ which can be extended in at least $(1 + o_k(1))2^{-k}N$ ways to a monochromatic $K_{k+1}$. This result is asymptotically best possible, as may be seen by considering a random colouring. Equivalently, defining the book $B_n^{(k)}$ to be the graph consisting of $n$ copies of $K_{k+1}$ all sharing a common $K_k$, we show that the Ramsey number $r(B_n^{(k)}) = 2^k n + o_k(n)$. In this form, our result answers a question of Erdős, Faudree, Rousseau and Schelp and establishes an asymptotic version of a conjecture of Thomason.

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arXiv2019

Books versus triangles at the extremal density

A celebrated result of Mantel shows that every graph on $n$ vertices with $\lfloor n^2/4 \rfloor + 1$ edges must contain a triangle. A robust version of this result, due to Rademacher, says that there must in fact be at least $\lfloor n/2 \rfloor$ triangles in any such graph. Another strengthening, due to the combined efforts of many authors starting with Erdős, says that any such graph must have an edge which is contained in at least $n/6$ triangles. Following Mubayi, we study the interplay between these two results, that is, between the number of triangles in such graphs and their book number, the largest number of triangles sharing an edge. Among other results, Mubayi showed that for any $1/6 \leq β< 1/4$ there is $γ> 0$ such that any graph on $n$ vertices with at least $\lfloor n^2/4\rfloor + 1$ edges and book number at most $βn$ contains at least $(γ-o(1))n^3$ triangles. He also asked for a more precise estimate for $γ$ in terms of $β$. We make a conjecture about this dependency and prove this conjecture for $β= 1/6$ and for $0.2495 \leq β< 1/4$, thereby answering Mubayi's question in these ranges.

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arXiv2019

Graphs with few paths of prescribed length between any two vertices

We use a variant of Bukh's random algebraic method to show that for every natural number $k \geq 2$ there exists a natural number $\ell$ such that, for every $n$, there is a graph with $n$ vertices and $Ω_k(n^{1 + 1/k})$ edges with at most $\ell$ paths of length $k$ between any two vertices. A result of Faudree and Simonovits shows that the bound on the number of edges is tight up to the implied constant.

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arXiv2019

Hypergraph cuts above the average

An r-cut of a k-uniform hypergraph H is a partition of the vertex set of H into r parts and the size of the cut is the number of edges which have a vertex in each part. A classical result of Edwards says that every m-edge graph has a 2-cut of size $m/2 + Ω(\sqrt{m})$, and this is best possible. That is, there exist cuts which exceed the expected size of a random cut by some multiple of the standard deviation. We study analogues of this and related results in hypergraphs. First, we observe that similarly to graphs, every m-edge k-uniform hypergraph has an r-cut whose size is $Ω(\sqrt m)$ larger than the expected size of a random r-cut. Moreover, in the case where k=3 and r=2 this bound is best possible and is attained by Steiner triple systems. Surprisingly, for all other cases (that is, if $k \geq 4$ or $r \geq 3$), we show that every m-edge k-uniform hypergraph has an r-cut whose size is $Ω(m^{5/9})$ larger than the expected size of a random r-cut. This is a significant difference in behaviour, since the amount by which the size of the largest cut exceeds the expected size of a random cut is now considerably larger than the standard deviation.

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arXiv2019

Hypergraph expanders from Cayley graphs

We present a simple mechanism, which can be randomised, for constructing sparse $3$-uniform hypergraphs with strong expansion properties. These hypergraphs are constructed using Cayley graphs over $\mathbb{Z}_2^t$ and have vertex degree which is polylogarithmic in the number of vertices. Their expansion properties, which are derived from the underlying Cayley graphs, include analogues of vertex and edge expansion in graphs, rapid mixing of the random walk on the edges of the skeleton graph, uniform distribution of edges on large vertex subsets and the geometric overlap property.

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arXiv2019

On the extremal number of subdivisions

One of the cornerstones of extremal graph theory is a result of Füredi, later reproved and given due prominence by Alon, Krivelevich and Sudakov, saying that if $H$ is a bipartite graph with maximum degree $r$ on one side, then there is a constant $C$ such that every graph with $n$ vertices and $C n^{2 - 1/r}$ edges contains a copy of $H$. This result is tight up to the constant when $H$ contains a copy of $K_{r,s}$ with $s$ sufficiently large in terms of $r$. We conjecture that this is essentially the only situation in which Füredi's result can be tight and prove this conjecture for $r = 2$. More precisely, we show that if $H$ is a $C_4$-free bipartite graph with maximum degree $2$ on one side, then there are positive constants $C$ and $δ$ such that every graph with $n$ vertices and $C n^{3/2 - δ}$ edges contains a copy of $H$. This answers a question of Erdős from 1988. The proof relies on a novel variant of the dependent random choice technique which may be of independent interest.

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arXiv2019

Independent arithmetic progressions

We show that there is a positive constant $c$ such that any graph on vertex set $[n]$ with at most $c n^2/k^2 \log k$ edges contains an independent set of order $k$ whose vertices form an arithmetic progression. We also present applications of this result to several questions in Ramsey theory.

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arXiv2018

Tower-type bounds for unavoidable patterns in words

A word $w$ is said to contain the pattern $P$ if there is a way to substitute a nonempty word for each letter in $P$ so that the resulting word is a subword of $w$. Bean, Ehrenfeucht and McNulty and, independently, Zimin characterised the patterns $P$ which are unavoidable, in the sense that any sufficiently long word over a fixed alphabet contains $P$. Zimin's characterisation says that a pattern is unavoidable if and only if it is contained in a Zimin word, where the Zimin words are defined by $Z_1 = x_1$ and $Z_n=Z_{n-1} x_n Z_{n-1}$. We study the quantitative aspects of this theorem, obtaining essentially tight tower-type bounds for the function $f(n,q)$, the least integer such that any word of length $f(n, q)$ over an alphabet of size $q$ contains $Z_n$. When $n = 3$, the first non-trivial case, we determine $f(n,q)$ up to a constant factor, showing that $f(3,q) = Θ(2^q q!)$.

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arXiv2018

Online Ramsey Numbers and the Subgraph Query Problem

The $(m,n)$-online Ramsey game is a combinatorial game between two players, Builder and Painter. Starting from an infinite set of isolated vertices, Builder draws an edge on each turn and Painter immediately paints it red or blue. Builder's goal is to force Painter to create either a red $K_m$ or a blue $K_n$ using as few turns as possible. The online Ramsey number $\tilde{r}(m,n)$ is the minimum number of edges Builder needs to guarantee a win in the $(m,n)$-online Ramsey game. By analyzing the special case where Painter plays randomly, we obtain an exponential improvement \[ \tilde{r}(n,n) \ge 2^{(2-\sqrt{2})n + O(1)} \] for the lower bound on the diagonal online Ramsey number, as well as a corresponding improvement \[ \tilde{r}(m,n) \ge n^{(2-\sqrt{2})m + O(1)} \] for the off-diagonal case, where $m\ge 3$ is fixed and $n\rightarrow\infty$. Using a different randomized Painter strategy, we prove that $\tilde{r}(3,n)=\tildeΘ(n^3)$, determining this function up to a polylogarithmic factor. We also improve the upper bound in the off-diagonal case for $m \geq 4$. In connection with the online Ramsey game with a random Painter, we study the problem of finding a copy of a target graph $H$ in a sufficiently large unknown Erdős--Rényi random graph $G(N,p)$ using as few queries as possible, where each query reveals whether or not a particular pair of vertices are adjacent. We call this problem the Subgraph Query Problem. We determine the order of the number of queries needed for complete graphs up to five vertices and prove general bounds for this problem.

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arXiv2018

Intervals in the Hales-Jewett theorem

The Hales-Jewett theorem states that for any $m$ and $r$ there exists an $n$ such that any $r$-colouring of the elements of $[m]^n$ contains a monochromatic combinatorial line. We study the structure of the wildcard set $S \subseteq [n]$ which determines this monochromatic line, showing that when $r$ is odd there are $r$-colourings of $[3]^n$ where the wildcard set of a monochromatic line cannot be the union of fewer than $r$ intervals. This is tight, as for $n$ sufficiently large there are always monochromatic lines whose wildcard set is the union of at most $r$ intervals.

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arXiv2018

Sidorenko's conjecture for higher tree decompositions

This is a companion note to our paper 'Some advances on Sidorenko's conjecture', elaborating on a remark in that paper that the approach which proves Sidorenko's conjecture for strongly tree-decomposable graphs may be extended to a broader class, comparable to that given in work of Szegedy, through further iteration.

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arXiv2018

Some advances on Sidorenko's conjecture

A bipartite graph $H$ is said to have Sidorenko's property if the probability that the uniform random mapping from $V(H)$ to the vertex set of any graph $G$ is a homomorphism is at least the product over all edges in $H$ of the probability that the edge is mapped to an edge of $G$. In this paper, we provide three distinct families of bipartite graphs that have Sidorenko's property. First, using branching random walks, we develop an embedding algorithm which allows us to prove that bipartite graphs admitting a certain type of tree decomposition have Sidorenko's property. Second, we use the concept of locally dense graphs to prove that subdivisions of certain graphs, including cliques, have Sidorenko's property. Third, we prove that if $H$ has Sidorenko's property, then the Cartesian product of $H$ with an even cycle also has Sidorenko's property.

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arXiv2018

Lines in Euclidean Ramsey theory

Let $\ell_m$ be a sequence of $m$ points on a line with consecutive points of distance one. For every natural number $n$, we prove the existence of a red/blue-coloring of $\mathbb{E}^n$ containing no red copy of $\ell_2$ and no blue copy of $\ell_m$ for any $m \geq 2^{cn}$. This is best possible up to the constant $c$ in the exponent. It also answers a question of Erdős, Graham, Montgomery, Rothschild, Spencer and Straus from 1973. They asked if, for every natural number $n$, there is a set $K \subset \mathbb{E}^1$ and a red/blue-coloring of $\mathbb{E}^n$ containing no red copy of $\ell_2$ and no blue copy of $K$.

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arXiv2018

The Green-Tao theorem: an exposition

The celebrated Green-Tao theorem states that the prime numbers contain arbitrarily long arithmetic progressions. We give an exposition of the proof, incorporating several simplifications that have been discovered since the original paper.

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