David Conlon

On off-diagonal hypergraph Ramsey numbers

David Conlon, Jacob Fox, Benjamin Gunby +4

A fundamental problem in Ramsey theory is to determine the growth rate in terms of $n$ of the Ramsey number $r (H, K_{n}^{(3)})$ of a fixed $3$ -uniform hypergraph $H$ versus the compl…

Subset sums, completeness and colorings

David Conlon, Jacob Fox, Huy Tuan Pham

We develop novel techniques which allow us to prove a diverse range of results relating to subset sums and complete sequences of positive integers, including solutions to several l…

math.CO2026

A note on arithmetic progressions with restricted differences

David Conlon, Jacob Fox, Huy Tuan Pham

In this note, we show how to adapt Tao's slice rank method to extend the Ellenberg--Gijswijt theorem on cap sets to the problem of forbidding arithmetic progressions with restricte…

Monochromatic combinatorial lines of length three

David Conlon

We show that there is a constant $c$ such that any colouring of the cube $[3]^{n}$ in $c lo g lo g n$ colours contains a monochromatic combinatorial line.

More on the extremal number of subdivisions

David Conlon, Oliver Janzer, Joonkyung Lee

Given a graph $H$ , the extremal number $ex (n, H)$ is the largest number of edges in an $H$ -free graph on $n$ vertices. We make progress on a number of conjectures about the…

Hypergraphs accumulate infinitely often

David Conlon, Bjarne SchÃ¼lke

We show that the set $I^^{(k)}$ of TurÃ¡n densities of $k$ -uniform hypergraphs has infinitely many accumulation points in $[0, 1)$ for every $k \geq 3$ . This extends an earlier resu…

Ramsey numbers and the Zarankiewicz problem

David Conlon, Sam Mattheus, Dhruv Mubayi +1

Building on recent work of Mattheus and VerstraÃ«te, we establish a general connection between Ramsey numbers of the form $r (F, t)$ for $F$ a fixed graph and a variant of the Zarank…

Hypergraphs accumulate

David Conlon, Bjarne SchÃ¼lke

We show that for every integer $k \geq 3$ , the set of TurÃ¡n densities of $k$ -uniform hypergraphs has an accumulation point in $[0, 1)$ . In particular, $1/2$ is an accumulation point…

The Ramsey number of books

David Conlon

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 monoch…

Cycle packing

David Conlon, Jacob Fox, Benny Sudakov

In the 1960s, ErdÅs and Gallai conjectured that the edge set of every graph on n vertices can be partitioned into O(n) cycles and edges. They observed that one can easily get an O…

Monochromatic cycle partitions in local edge colourings

David Conlon, Maya Stein

An edge colouring of a graph is said to be an $r$ -local colouring if the edges incident to any vertex are coloured with at most $r$ colours. Generalising a result of Bessy and Thom…

math.NT2014

A relative SzemerÃ©di theorem

David Conlon, Jacob Fox, Yufei Zhao

The celebrated Green-Tao theorem states that there are arbitrarily long arithmetic progressions in the primes. One of the main ingredients in their proof is a relative SzemerÃ©di t…

Set-coloring Ramsey numbers and error-correcting codes near the zero-rate threshold

David Conlon, Jacob Fox, Huy Tuan Pham +1

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 $s$ c…

Hereditary quasirandomness without regularity

David Conlon, Jacob Fox, Benny Sudakov

A result of Simonovits and SÃ³s states that for any fixed graph $H$ and any $I^µ > 0$ there exists $I^´ > 0$ such that if $G$ is an $n$ -vertex graph with the property that every $S \su…

Short proofs of some extremal results II

David Conlon, Jacob Fox, Benny Sudakov

We prove several results from different areas of extremal combinatorics, including complete or partial solutions to a number of open problems. These results, coming mainly from ext…

Hypergraph expanders of all uniformities from Cayley graphs

David Conlon, Jonathan Tidor, Yufei Zhao

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, o…

When are off-diagonal hypergraph Ramsey numbers polynomial?

David Conlon, Jacob Fox, Benjamin Gunby +5

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 substan…

math.CO2007

A note on lower bounds for hypergraph Ramsey numbers

David Conlon

We improve upon the lower bound for 3-colour hypergraph Ramsey numbers, showing, in the 3-uniform case, that \[r_3 (l,l,l) \geq 2^{l^{c \log \log l}}.\] The old bound, due to ErdÅ…

Graphs with few paths of prescribed length between any two vertices

David Conlon

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 $ℓ$ such that, for every $n$ , there is a graph…

A sequence of triangle-free pseudorandom graphs

David Conlon

A construction of Alon yields a sequence of highly pseudorandom triangle-free graphs with edge density significantly higher than one might expect from comparison with random graphs…

Extremal results in sparse pseudorandom graphs

David Conlon, Jacob Fox, Yufei Zhao

SzemerÃ©di's regularity lemma is a fundamental tool in extremal combinatorics. However, the original version is only helpful in studying dense graphs. In the 1990s, Kohayakawa and…

Ramsey-type results for semi-algebraic relations

David Conlon, Jacob Fox, JÃ¡nos Pach +2

A k-ary semi-algebraic relation E on R^d is a subset of R^{kd}, the set of k-tuples of points in R^d, which is determined by a finite number of polynomial equations and inequalitie…

math.CO2026

Two counterexamples to a conjecture about even cycles

David Conlon, Eion Mulrenin, Cosmin Pohoata

A conjecture of VerstraÃ«te states that for any fixed $ℓ < k$ there exists a positive constant $c$ such that any $C_{2 k}$ -free graph $G$ contains a $C_{2 ℓ}$ -free subgraph wit…

A question of ErdÅs and Graham on Egyptian fractions

David Conlon, Jacob Fox, Xiaoyu He +4

Answering a question of ErdÅs 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 int…

Short proofs of some extremal results III

David Conlon, Jacob Fox, Benny Sudakov

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 f…

Some remarks on the Zarankiewicz problem

David Conlon

The Zarankiewicz problem asks for an estimate on $z (m, n; s, t)$ , the largest number of $1$ 's in an $m \times n$ matrix with all entries $0$ or $1$ containing no $s \times t$ subma…

Intervals in the Hales-Jewett theorem

David Conlon, Nina Kamcev

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…

The ErdÅs-GyÃ¡rfÃ¡s problem on generalized Ramsey numbers

David Conlon, Jacob Fox, Choongbum Lee +1

Fix positive integers $p$ and $q$ with $2 \leq q \leq (2 p)$ . An edge-coloring of the complete graph $K_{n}$ is said to be a $(p, q)$ -coloring if every $K_{p}$ receives at leas…

math.CO2017

Quasirandom Cayley graphs

David Conlon, Yufei Zhao

We prove that the properties of having small discrepancy and having small second eigenvalue are equivalent in Cayley graphs, extending a result of Kohayakawa, RÃ¶dl, and Schacht, w…

Tower-type bounds for unavoidable patterns in words

David Conlon, Jacob Fox, Benny Sudakov

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, Ehrenfeuc…

math.CO2008

Hypergraph Ramsey numbers

David Conlon, Jacob Fox, Benny Sudakov

The Ramsey number r_k(s,n) is the minimum N such that every red-blue coloring of the k-tuples of an N-element set contains either a red set of size s or a blue set of size n, where…

Recent developments in graph Ramsey theory

David Conlon, Jacob Fox, Benny Sudakov

Given a graph $H$ , the Ramsey number $r (H)$ is the smallest natural number $N$ such that any two-colouring of the edges of $K_{N}$ contains a monochromatic copy of $H$ . The existence…

Ramsey numbers of cubes versus cliques

David Conlon, Jacob Fox, Choongbum Lee +1

The cube graph Q_n is the skeleton of the n-dimensional cube. It is an n-regular graph on 2^n vertices. The Ramsey number r(Q_n, K_s) is the minimum N such that every graph of orde…

Non-spherical sets versus lines in Euclidean Ramsey theory

David Conlon, Jakob FÃ¼hrer

We show that for every non-spherical set $X$ in $E^{d}$ , there exists a natural number $m$ and a red/blue-colouring of $E^{n}$ for every $n$ such that there is no red…

We show that if $I^»_{1}, \dots, I^»_{k}$ are algebraic numbers, then $∣ A + I^»_{1} \cdot A + \dots + I^»_{k} \cdot A ∣ \geq H (I^»_{1}, \dots, I^»_{k}) ∣ A ∣ - o (∣ A ∣)$ for all finite subsets $A$ of $C$ , w…

Domination inequalities and dominating graphs

David Conlon, Joonkyung Lee

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…

Repeated patterns in proper colourings

David Conlon, Mykhaylo Tyomkyn

For a fixed graph $H$ , what is the smallest number of colours $C$ such that there is a proper edge-colouring of the complete graph $K_{n}$ with $C$ colours containing no two vertex-d…

Simplicial TurÃ¡n problems

David Conlon, SimÃ³n Piga, Bjarne SchÃ¼lke

A simplicial complex $H$ consists of a pair of sets $(V, E)$ where $V$ is a set of vertices and $E \subseteq P (V)$ is a collection of subsets of $V$ closed under taking subs…

On-line Ramsey numbers

David Conlon

Consider the following game between two players, Builder and Painter. Builder draws edges one at a time and Painter colours them, in either red or blue, as each appears. Builder's…

Fixing a hole

David Conlon, Jeck Lim

We show that any finite $S \subset R^{d}$ in general position has arbitrarily large supersets $T \supseteq S$ in general position with the property that $T$ contains no empt…

math.CO2006

A New Upper Bound for Diagonal Ramsey Numbers

David Conlon

We prove a new upper bound for diagonal two-colour Ramsey numbers, showing that there exists a constant $C$ such that \[r(k+1, k+1) \leq k^{- C \frac{\log k}{\log \log k}} \binom{2…

math.CO2026

Combinatorial theorems relative to sparse sets

David Conlon

A key theme in modern extremal combinatorics is the study of classical combinatorial theorems relative to sparse subsets of their natural settings. Here we describe some of the rec…

The upper logarithmic density of monochromatic subset sums

David Conlon, Jacob Fox, Huy Tuan Pham

We show that in any two-coloring of the positive integers there is a color for which the set of positive integers that can be represented as a sum of distinct elements with this co…

math.CO2012

Graph removal lemmas

David Conlon, Jacob Fox

The graph removal lemma states that any graph on n vertices with o(n^{v(H)}) copies of a fixed graph H may be made H-free by removing o(n^2) edges. Despite its innocent appearance,…

Large almost monochromatic subsets in hypergraphs

David Conlon, Jacob Fox, Benny Sudakov

We show that for all $ℓ$ and $I^µ > 0$ there is a constant $c = c (ℓ, I^µ) > 0$ such that every $ℓ$ -coloring of the triples of an $N$ -element set contains a subset $S$ of size $c\sq…

Set-coloring Ramsey numbers via codes

David Conlon, Jacob Fox, Xiaoyu He +3

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 $s$ c…

Random multilinear maps and the ErdÅs box problem

David Conlon, Cosmin Pohoata, Dmitriy Zakharov

By using random multilinear maps, we provide new lower bounds for the ErdÅs box problem, the problem of estimating the extremal number of the complete $d$ -partite $d$ -uniform hype…

Threshold Ramsey multiplicity for paths and even cycles

David Conlon, Jacob Fox, Benny Sudakov +1

The Ramsey number $r (H)$ of a graph $H$ is the minimum integer $n$ such that any two-coloring of the edges of the complete graph $K_{n}$ contains a monochromatic copy of $H$ . While t…

Distinct volume subsets

David Conlon, Jacob Fox, William Gasarch +3

Suppose that $a$ and $d$ are positive integers with $a \geq 2$ . Let $h_{a, d} (n)$ be the largest integer $t$ such that any set of $n$ points in $R^{d}$ contains a subset of $…

Hedgehogs are not colour blind

David Conlon, Jacob Fox, VojtÄch RÃ¶dl

We exhibit a family of $3$ -uniform hypergraphs with the property that their $2$ -colour Ramsey numbers grow polynomially in the number of vertices, while their $4$ -colour Ramsey num…

Around the positive graph conjecture

David Conlon, Joonkyung Lee, Leo Versteegen

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 su…

Independent arithmetic progressions

David Conlon, Jacob Fox, Benny Sudakov

We show that there is a positive constant $c$ such that any graph on vertex set $[n]$ with at most $c n^{2} / k^{2} lo g k$ edges contains an independent set of order $k$ whose vertices…

math.CO2017

Finite reflection groups and graph norms

David Conlon, Joonkyung Lee

Given a graph $H$ on vertex set ${1, 2, \dots, n}$ and a function $f : [0, 1]^{2} \to R$ , define \begin{align*} \|f\|_{H}:=\left\vert\int \prod_{ij\in E(H)}f(x_i,x_j)d…

math.NT2013

Linear forms from the Gowers uniformity norm

David Conlon, Jacob Fox, Yufei Zhao

This is a companion note to our paper 'A relative SzemerÃ©di theorem', elaborating on a concluding remark. In that paper, we showed how to prove a relative SzemerÃ©di theorem for $…

Ramsey numbers of trails and circuits

David Conlon, Mykhaylo Tyomkyn

We show that every two-colouring of the edges of the complete graph $K_{n}$ contains a monochromatic trail or circuit of length at least $2 n^{2} /9 + o (n^{2})$ , which is asymptotically bes…

math.CO2010

An extremal theorem in the hypercube

David Conlon

The hypercube Q_n is the graph whose vertex set is {0,1}^n and where two vertices are adjacent if they differ in exactly one coordinate. For any subgraph H of the cube, let ex(Q_n,…

Sidorenko's conjecture for blow-ups

David Conlon, Joonkyung Lee

A celebrated conjecture of Sidorenko and ErdÅs-Simonovits states that, for all bipartite graphs $H$ , quasirandom graphs contain asymptotically the minimum number of copies of $H$ …

The regularity method for graphs with few 4-cycles

David Conlon, Jacob Fox, Benny Sudakov +1

We develop a sparse graph regularity method that applies to graphs with few 4-cycles, including new counting and removal lemmas for 5-cycles in such graphs. Some applications inclu…

Short proofs of some extremal results

David Conlon, Jacob Fox, Benny Sudakov

We prove several results from different areas of extremal combinatorics, giving complete or partial solutions to a number of open problems. These results, coming from areas such as…

math.NT2026

Simultaneous popular polynomial differences over finite fields

David Conlon, Dingding Dong, Guo-Dong Hong

Green's popular difference theorem says that for every $\varepsilon>0$, all sufficiently large primes $p$, and every set $A\subseteq\mathbb F_p$ of density $Î±$, there exis…

Almost-spanning universality in random graphs

David Conlon, Asaf Ferber, Rajko Nenadov +1

A graph $G$ is said to be $H (n, I^)$ -universal if it contains every graph on $n$ vertices with maximum degree at most $I^$ . It is known that for any $ε > 0$ and a…

math.CO2007

On the Ramsey multiplicity of complete graphs

David Conlon

We show that, for $n$ large, there must exist at least \[\frac{n^t}{C^{(1+o(1))t^2}}\] monochromatic $K_{t}$ s in any two-colouring of the edges of $K_{n}$ , where $C \approx 2.18$ is an…

Sums of transcendental dilates

David Conlon, Jeck Lim

We show that there is an absolute constant $c > 0$ such that $∣ A + I^» \cdot A ∣ \geq e^{c l o g ∣ A ∣} ∣ A ∣$ for any finite subset $A$ of $R$ and any transcendental number $Î»\i…

Ramsey games near the critical threshold

David Conlon, Shagnik Das, Joonkyung Lee +1

A well-known result of RÃ¶dl and RuciÅski 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.…

math.NT2018

The Green-Tao theorem: an exposition

David Conlon, Jacob Fox, Yufei Zhao

The celebrated Green-Tao theorem states that the prime numbers contain arbitrarily long arithmetic progressions. We give an exposition of the proof, incorporating several simplific…

math.CO2011

Bounds for graph regularity and removal lemmas

David Conlon, Jacob Fox

We show, for any positive integer k, that there exists a graph in which any equitable partition of its vertices into k parts has at least ck^2/\log^* k pairs of parts which are not…

Monochromatic components with many edges

David Conlon, Sammy Luo, Mykhaylo Tyomkyn

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…

On norming systems of linear equations

Seokjoon Cho, David Conlon, Joonkyung Lee +2

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 n…

Rational exponents near two

David Conlon, Oliver Janzer

A longstanding conjecture of ErdÅs and Simonovits states that for every rational $r$ between $1$ and $2$ there is a graph $H$ such that the largest number of edges in an $H$ -free…

Sums of linear transformations

David Conlon, Jeck Lim

We show that if $L_{1}$ and $L_{2}$ are linear transformations from $Z^{d}$ to $Z^{d}$ satisfying certain mild conditions, then, for any finite subset…

Homogeneous structures in subset sums and non-averaging sets

David Conlon, Jacob Fox, Huy Tuan Pham

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 so…

On the size-Ramsey number of grids

David Conlon, Rajko Nenadov, MiloÅ¡ TrujiÄ

We show that the size-Ramsey number of the $n \times n$ grid graph is $O (n^{5/4})$ , improving a previous bound of $n^{3/2 + o (1)}$ by Clemens, Miralaei, Reding, Schac…

The Ramsey number of dense graphs

David Conlon

The Ramsey number r(H) of a graph H is the smallest number n such that, in any two-colouring of the edges of K_n, there is a monochromatic copy of H. We study the Ramsey number of…

math.CO2007

Ramsey numbers of sparse hypergraphs

David Conlon, Jacob Fox, Benny Sudakov

We give a short proof that any k-uniform hypergraph H on n vertices with bounded degree Îhas Ramsey number at most c(Î, k)n, for an appropriate constant c(Î, k). This result was…

On the grid Ramsey problem and related questions

David Conlon, Jacob Fox, Choongbum Lee +1

The Hales--Jewett theorem is one of the pillars of Ramsey theory, from which many other results follow. A celebrated theorem of Shelah says that Hales--Jewett numbers are primitive…

The size-Ramsey number of cubic graphs

David Conlon, Rajko Nenadov, MiloÅ¡ TrujiÄ

We show that the size-Ramsey number of any cubic graph with $n$ vertices is $O (n^{8/5})$ , improving a bound of $n^{5/3 + o (1)}$ due to Kohayakawa, RÃ¶dl, Schacht, and SzemerÃ©di. T…

A note on induced Ramsey numbers

David Conlon, Domingos Dellamonica, Steven La Fleur +2

The induced Ramsey number $r_{ind} (F)$ of a $k$ -uniform hypergraph $F$ is the smallest natural number $n$ for which there exists a $k$ -uniform hypergraph $G$ on $n$ vertic…

Combinatorial theorems relative to a random set

David Conlon

We describe recent advances in the study of random analogues of combinatorial theorems.

An improved bound for the stepping-up lemma

David Conlon, Jacob Fox, Benny Sudakov

The partition relation N \to (n)_{\ell}^k means that whenever the k-tuples of an N-element set are \ell-colored, there is a monochromatic set of size n, where a set is called monoc…

Books versus triangles at the extremal density

David Conlon, Jacob Fox, Benny Sudakov

A celebrated result of Mantel shows that every graph on $n$ vertices with $⌊ n^{2} /4 ⌋ + 1$ edges must contain a triangle. A robust version of this result, due to Rademac…

Lower bounds for multicolor Ramsey numbers

David Conlon, Asaf Ferber

We give an exponential improvement to the lower bound on diagonal Ramsey numbers for any fixed number of colors greater than two.

Three early problems on size Ramsey numbers

David Conlon, Jacob Fox, Yuval Wigderson

The size Ramsey number of a graph $H$ is defined as the minimum number of edges in a graph $G$ such that there is a monochromatic copy of $H$ in every two-coloring of $E (G)$ . The s…

Threshold Ramsey multiplicity for odd cycles

David Conlon, Jacob Fox, Benny Sudakov +1

The Ramsey number $r (H)$ of a graph $H$ is the minimum $n$ such that any two-coloring of the edges of the complete graph $K_{n}$ contains a monochromatic copy of $H$ . The threshold R…

On the extremal number of subdivisions

David Conlon, Joonkyung Lee

One of the cornerstones of extremal graph theory is a result of FÃ¼redi, later reproved and given due prominence by Alon, Krivelevich and Sudakov, saying that if $H$ is a bipartite…

More on lines in Euclidean Ramsey theory

David Conlon, Yu-Han Wu

Let $ℓ_{m}$ be a sequence of $m$ points on a line with consecutive points at distance one. Answering a question raised by Fox and the first author and independently by Arman and T…

A New Bound for the Brown--ErdÅs--SÃ³s Problem

David Conlon, Lior Gishboliner, Yevgeny Levanzov +1

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…

math.CO2017

Rational exponents in extremal graph theory

Boris Bukh, David Conlon

Given a family of graphs $H$ , the extremal number $ex (n, H)$ is the largest $m$ for which there exists a graph with $n$ vertices and $m$ edges containi…

Hypergraph Ramsey numbers of cliques versus stars

David Conlon, Jacob Fox, Xiaoyu He +3

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 $(2 n)$ edges incid…

Some advances on Sidorenko's conjecture

David Conlon, Jeong Han Kim, Choongbum Lee +1

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…

Off-diagonal book Ramsey numbers

David Conlon, Jacob Fox, Yuval Wigderson

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^{(…

Ramsey numbers of books and quasirandomness

David Conlon, Jacob Fox, Yuval Wigderson

The book graph $B_{n}^{(k)}$ consists of $n$ copies of $K_{k + 1}$ joined along a common $K_{k}$ . The Ramsey numbers of $B_{n}^{(k)}$ are known to have strong connections to the classical…

math.CO2010

An approximate version of Sidorenko's conjecture

David Conlon, Jacob Fox, Benny Sudakov

A beautiful conjecture of ErdÅs-Simonovits and Sidorenko states that if H is a bipartite graph, then the random graph with edge density p has in expectation asymptotically the min…

Difference sets in $R^{d}$

David Conlon, Jeck Lim

Let $d \geq 2$ be a natural number. We show that $∣ A - A ∣ \geq (2 d - 2 + \frac{1}{d - 1}) ∣ A ∣ - (2 d^{2} - 4 d + 3)$ for any sufficiently large finite subset $A$ of $R^{d}$ that…

Extremal numbers of cycles revisited

David Conlon

We give a simple geometric interpretation of an algebraic construction of Wenger that yields $n$ -vertex graphs with no cycle of length $4$ , $6$ or $10$ and close to the maximum num…

Hypergraph expanders from Cayley graphs

David Conlon

We present a simple mechanism, which can be randomised, for constructing sparse $3$ -uniform hypergraphs with strong expansion properties. These hypergraphs are constructed using Ca…

Lines in Euclidean Ramsey theory

David Conlon, Jacob Fox

Let $ℓ_{m}$ 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{…

Ordered Ramsey numbers

David Conlon, Jacob Fox, Choongbum Lee +1

Given a labeled graph $H$ with vertex set ${1, 2, \dots, n}$ , the ordered Ramsey number $r_{<} (H)$ is the minimum $N$ such that every two-coloring of the edges of the complete graph…

Everywhere unbalanced configurations

David Conlon, Jeck Lim

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…

Even cycles in graphs avoiding longer even cycles

David Conlon, Eion Mulrenin, Cosmin Pohoata

A conjecture of VerstraÃ«te states that for any fixed $ℓ < k$ there exists a positive constant $c$ such that any $C_{2 k}$ -free graph $G$ contains a $C_{2 ℓ}$ -free subgraph wit…