David Conlon
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View article: Non-spherical sets versus lines in Euclidean Ramsey theory
Non-spherical sets versus lines in Euclidean Ramsey theory Open
We show that for every non-spherical set X in $\mathbb {E}^d$ , there exists a natural number m and a red/blue-coloring of $\mathbb {E}^n$ for every n such that there is no red copy of X and no blue progression of length m with each consec…
View article: Hypergraphs accumulate infinitely often
Hypergraphs accumulate infinitely often Open
We show that the set $Π^{(k)}$ of Turán densities of $k$-uniform hypergraphs has infinitely many accumulation points in $[0,1)$ for every $k \geq 3$. This extends an earlier result of ours showing that $Π^{(k)}$ has at least one such accum…
View article: Even cycles in graphs avoiding longer even cycles
Even cycles in graphs avoiding longer even cycles Open
A conjecture of Verstraëte states that for any fixed $\ell < k$ there exists a positive constant $c$ such that any $C_{2k}$-free graph $G$ contains a $C_{2\ell}$-free subgraph with at least $c |E(G)|$ edges. For $\ell = 2$, this conjecture…
View article: On the extremal number of incidence graphs
On the extremal number of incidence graphs Open
Given a graph $H$ and a natural number $n$, the extremal number $\mathrm{ex}(n, H)$ is the largest number of edges in an $n$-vertex graph containing no copy of $H$. In this paper, we obtain a general upper bound for the extremal number of …
View article: On the clique number of random Cayley graphs and related topics
On the clique number of random Cayley graphs and related topics Open
We prove that a random Cayley graph on a group of order $N$ has clique number $O(\log N \log \log N)$ with high probability. This bound is best possible up to the constant factor for certain groups, including~$\mathbb{F}_2^n$, and improves…
View article: On norming systems of linear equations
On norming systems of linear equations Open
A system of linear equations $L$ is said to be norming if a natural functional $t_L(\cdot)$ giving a weighted count for the set of solutions to the system can be used to define a norm on the space of real-valued functions on $\mathbb{F}_q^…
View article: When are off-diagonal hypergraph Ramsey numbers polynomial?
When are off-diagonal hypergraph Ramsey numbers polynomial? Open
A natural open problem in Ramsey theory is to determine those $3$-graphs $H$ for which the off-diagonal Ramsey number $r(H, K_n^{(3)})$ grows polynomially with $n$. We make substantial progress on this question by showing that if $H$ is ti…
View article: Sums of dilates over groups of prime order
Sums of dilates over groups of prime order Open
For $p$ prime, $A \subseteq \mathbb{Z}/p\mathbb{Z}$ and $λ\in \mathbb{Z}$, the sum of dilates $A + λ\cdot A$ is defined by \[A + λ\cdot A = \{a + λa' : a, a' \in A\}.\] The basic problem on such sums of dilates asks for the minimum size of…
View article: Non-spherical sets versus lines in Euclidean Ramsey theory
Non-spherical sets versus lines in Euclidean Ramsey theory Open
We show that for every non-spherical set $X$ in $\mathbb{E}^d$, there exists a natural number $m$ and a red/blue-colouring of $\mathbb{E}^n$ for every $n$ such that there is no red copy of X and no blue progression of length $m$ with each …
View article: Hypergraphs accumulate
Hypergraphs accumulate Open
We show that for every integer $k\geq3$, the set of Turán densities of $k$-uniform hypergraphs has an accumulation point in $[0,1)$. In particular, $1/2$ is an accumulation point for the set of Turán densities of $3$-uniform hypergraphs.
View article: Big line or big convex polygon
Big line or big convex polygon Open
Let $ES_{\ell}(n)$ be the minimum $N$ such that every $N$-element point set in the plane contains either $\ell$ collinear members or $n$ points in convex position. We prove that there is a constant $C>0$ such that, for each $\ell, n \ge 3$…
View article: Around the positive graph conjecture
Around the positive graph conjecture Open
A graph $H$ is said to be positive if the homomorphism density $t_H(G)$ is non-negative for all weighted graphs $G$. The positive graph conjecture proposes a characterisation of such graphs, saying that a graph is positive if and only if i…
View article: A question of Erdős and Graham on Egyptian fractions
A question of Erdős and Graham on Egyptian fractions Open
Answering a question of Erdős and Graham, we show that for each fixed positive rational number $x$ the number of ways to write $x$ as a sum of reciprocals of distinct positive integers each at most $n$ is $2^{(c_x + o(1))n}$ for an explici…
View article: Monochromatic components with many edges
Monochromatic components with many edges Open
Given an r-edge-coloring of the complete graph K n , what is the largest number of edges in a monochromatic connected component?This natural question has only recently received the attention it deserves, with work by two disjoint subsets o…
View article: Homogeneous structures in subset sums and non-averaging sets
Homogeneous structures in subset sums and non-averaging sets Open
We show that for every positive integer $k$ there are positive constants $C$ and $c$ such that if $A$ is a subset of $\{1, 2, \dots, n\}$ of size at least $C n^{1/k}$, then, for some $d \leq k-1$, the set of subset sums of $A$ contains a h…
View article: Simplicial Turán problems
Simplicial Turán problems Open
A simplicial complex $H$ consists of a pair of sets $(V,E)$ where $V$ is a set of vertices and $E\subseteq\mathscr{P}(V)$ is a collection of subsets of $V$ closed under taking subsets. Given a simplicial complex $F$ and $n\in \mathbb N$, t…
View article: Everywhere unbalanced configurations
Everywhere unbalanced configurations Open
An old problem in discrete geometry, originating with Kupitz, asks whether there is a fixed natural number $k$ such that every finite set of points in the plane has a line through at least two of its points where the number of points on ei…
View article: Ramsey numbers and the Zarankiewicz problem
Ramsey numbers and the Zarankiewicz problem Open
Building on recent work of Mattheus and Verstraëte, we establish a general connection between Ramsey numbers of the form $r(F,t)$ for $F$ a fixed graph and a variant of the Zarankiewicz problem asking for the maximum number of 1s in an $m$…
View article: Extremal numbers and Sidorenko's conjecture
Extremal numbers and Sidorenko's conjecture Open
Sidorenko's conjecture states that, for all bipartite graphs $H$, quasirandom graphs contain asymptotically the minimum number of copies of $H$ taken over all graphs with the same order and edge density. While still open for graphs, the an…
View article: On the size-Ramsey number of grids
On the size-Ramsey number of grids Open
We show that the size-Ramsey number of the $\sqrt{n} \times \sqrt{n}$ grid graph is $O(n^{5/4})$ , improving a previous bound of $n^{3/2 + o(1)}$ by Clemens, Miralaei, Reding, Schacht, and Taraz.
View article: Set-coloring Ramsey numbers and error-correcting codes near the zero-rate threshold
Set-coloring Ramsey numbers and error-correcting codes near the zero-rate threshold Open
For positive integers $n,r,s$ with $r > s$, the set-coloring Ramsey number $R(n;r,s)$ is the minimum $N$ such that if every edge of the complete graph $K_N$ receives a set of $s$ colors from a palette of $r$ colors, then there is a subset …
View article: Hypergraph Ramsey numbers of cliques versus stars
Hypergraph Ramsey numbers of cliques versus stars Open
Let denote the complete 3‐uniform hypergraph on vertices and the 3‐uniform hypergraph on vertices consisting of all edges incident to a given vertex. Whereas many hypergraph Ramsey numbers grow either at most polynomially or at least expon…
View article: Domination inequalities and dominating graphs
Domination inequalities and dominating graphs Open
We say that a graph $H$ dominates another graph $H'$ if the number of homomorphisms from $H'$ to any graph $G$ is dominated, in an appropriate sense, by the number of homomorphisms from $H$ to $G$. We study the family of dominating graphs,…
View article: Off-diagonal book Ramsey numbers
Off-diagonal book Ramsey numbers Open
The book graph $B_n ^{(k)}$ consists of $n$ copies of $K_{k+1}$ joined along a common $K_k$ . In the prequel to this paper, we studied the diagonal Ramsey number $r(B_n ^{(k)}, B_n ^{(k)})$ . Here we consider the natural off-diagonal varia…
View article: Sums of transcendental dilates
Sums of transcendental dilates Open
We show that there is an absolute constant $c>0$ such that $|A+λ\cdot A|\geq e^{c\sqrt{\log |A|}}|A|$ for any finite subset $A$ of $\mathbb{R}$ and any transcendental number $λ\in\mathbb{R}$. By a construction of Konyagin and Laba, this is…
View article: Hypergraph Ramsey numbers of cliques versus stars
Hypergraph Ramsey numbers of cliques versus stars Open
Let $K_m^{(3)}$ denote the complete $3$-uniform hypergraph on $m$ vertices and $S_n^{(3)}$ the $3$-uniform hypergraph on $n+1$ vertices consisting of all $\binom{n}{2}$ edges incident to a given vertex. Whereas many hypergraph Ramsey numbe…
View article: A new bound for the Brown–Erdős–Sós problem
A new bound for the Brown–Erdős–Sós problem Open
Let f(n, v, e) denote the maximum number of edges in a 3-uniform hypergraph not containing e edges spanned by at most v vertices. One of the most influential open problems in extremal combinatorics then asks, for a given number of edges e …