arXiv subjects

David Conlon

Publications and source records attributed to David Conlon.

At least 19 recordsLinked to original sources

arXiv2026

Upper Bounds for CSP Non-redundancy via Locally Decodable Codes

In the study of constraint satisfaction problems (CSPs), the non-redundancy of a relation measures the maximum number of clauses in a CSP instance which can be logically independent. In the last few years, CSP non-redundancy has found a number of applications including in query complexity, sparsification, kernelization and streaming. In this paper, we present a novel technique for upper-bounding the non-redundancy of some relations using locally decodable codes (LDCs) and locally correctable codes (LCCs). More precisely, known limitations on the structure of 2-query LDCs and LCCs can be translated into limitations on the size of certain non-redundant CSP instances. This technique builds on a technique used by Brakensiek and Guruswami [STOC 2025] to bound non-redundancy via techniques from matching vector codes. Using LDCs and LCCs, we give improved upper bounds on non-redundancy of the BCK predicate pioneered by Bessiere, Carbonnel, and Katsirelos [AAAI 2020] as well as its well-studied Boolean variant. We also give the first example of a Boolean predicate whose non-redundancy is provably not of the form $n^{k\pm o(1)}$ for some positive integer $k$.

cs.DM↗

arXiv2026

A proof of the rational exponents conjecture

The rational exponents conjecture of Erdős and Simonovits asserts that for every rational $r\in [1,2]$ there exists a graph $H$ such that $\mathrm{ex}(n,H)=Θ(n^r)$. We present a proof of this conjecture claimed by Epoch AI, explaining how it builds upon ideas and methods developed by many researchers.

math.CO↗

arXiv2026

New bounds for off-diagonal hypergraph Ramsey numbers

We study a conjecture of Conlon, Fox, Gunby, He, Mubayi, Suk, Verstraëte, and Yu, which characterizes the $3$-graphs $H$ for which the off-diagonal Ramsey number $r(H,n)$ is polynomial. We prove the conjecture for $3$-graphs with at most five tight components. Additionally, we show that the full conjecture is equivalent to its specialization to linear $3$-graphs. We prove quasipolynomial upper bounds for a class of `forward-doubleable' $3$-graphs, which includes the Fano plane. Finally, we give an exponential lower bound for `non-forward-laminar' $3$-graphs and use this to completely classify the Ramsey growth of a natural family of 3-graphs called chains.

math.CO↗

arXiv2026

Combinatorial theorems relative to sparse sets

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 recent progress in this area and state a number of problems that remain open and pressing.

math.CO↗

arXiv2026

Simultaneous popular polynomial differences over finite fields

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 $α$, there exists a nonzero $d\in\mathbb F_p$ such that \[ \mathbb E_{x\in\mathbb F_p} 1_A(x)1_A(x+d)1_A(x+2d) \geq α^3-\varepsilon. \] We show that a stronger simultaneous popular difference phenomenon holds for polynomial configurations. Namely, if $\mathcal P=\{P_1,\dots,P_k\} \subset \mathbb Z[t]$ is a fixed collection of linearly independent polynomials with zero constant terms, we show that for every $\varepsilon>0$, all sufficiently large primes $p$, and every set $A\subseteq\mathbb F_p$ of density $α$, there exists a nonzero $d\in\mathbb F_p$ such that \[ \mathbb E_{x\in\mathbb F_p} 1_A(x) \prod_{i=1}^k 1_A\bigl(x+P_i(d)\bigr)^{ω_i} \geq α^{1+\sum_iω_i}-\varepsilon \] simultaneously for every $ω=(ω_1,\dots,ω_k)\in\{0,1\}^k$. We also show that such simultaneous popular difference phenomena have sharp limitations by proving that for every sufficiently large prime $p$, there is a constant $c>0$ such that, for all sufficiently large $n$, one can find a set $A\subseteq\mathbb F_p^n$ of density $1/2+o_n(1)$ satisfying \[ \max_{d\neq 0} \min\left\{ \mathbb E_{x\in\mathbb F_p^n} 1_A(x)1_A(x+d)1_A(x+2d), \mathbb E_{x\in\mathbb F_p^n} 1_A(x)1_A(x+2d)1_A(x+4d) \right\} \leq \frac18-c. \] That is, the strengthening of Green's result, in this case over $\mathbb F_p^n$ for $p$ fixed and $n$ tending to infinity, requiring that both $d$ and $2d$ are simultaneously popular differences for three-term arithmetic progressions is false.

math.NT↗

arXiv2026

A note on arithmetic progressions with restricted differences

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 restricted differences. In particular, we show that if $q$ is an odd prime power, there is $\varepsilon_q>0$ such that if $S \subseteq \mathbb{F}_q$ with $0 \in S$ and $|S|>(q+1)/2$ and $A \subseteq \mathbb{F}_q^n$ contains no three-term arithmetic progression whose common difference is in $S^n$, then $|A| \leq q^{(1-\varepsilon_q)n}$.

math.CO↗

arXiv2026

Two counterexamples to a conjecture about even cycles

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 was verified by Kühn and Osthus in 2004. We identify two counterexamples to this conjecture for $\ell = 4$ and $k=5$: the first comes from a recent construction of a dense $C_{10}$-free subgraph of the hypercube and the second from Wenger's construction for extremal $C_{10}$-free graphs.

math.CO↗

arXiv2025

A question of Erdős and Graham on Egyptian fractions

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 explicit constant $c_x$ increasing with $x$.

math.CO↗

arXiv2025

When are off-diagonal hypergraph Ramsey numbers polynomial?

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 tightly connected or has at most two tight components, then $r(H, K_n^{(3)})$ grows polynomially if and only if $H$ is contained in an iterated blowup of an edge.

math.CO↗

arXiv2025

Sums of algebraic dilates

We show that if $λ_1,\ldots,λ_k$ are algebraic numbers, then $$|A+λ_1\cdot A+\dots+λ_k\cdot A|\geq H(λ_1,\ldots,λ_k)|A|-o(|A|)$$ for all finite subsets $A$ of $\mathbb{C}$, where $H(λ_1,\ldots,λ_k)$ is an explicit constant that is best possible. The proof combines several ingredients, including a lower bound estimate on the measure of sums of linear transformations of compact sets in $\mathbb{R}^d$, a variant of Freiman's theorem tuned specifically to sums of dilates and the analysis of what we call lattice density, which succinctly captures how a subset of $\mathbb{Z}^d$ is arranged relative to a given flag of lattices. As an application, we revisit the study of sums of linear transformations of finite sets, in particular proving an asymptotically best possible lower bound for sums of two linear transformations.

math.CO↗

arXiv2025

Hypergraphs accumulate infinitely often

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 accumulation point.

math.CO↗

arXiv2025

Everywhere unbalanced configurations

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 either side of this line differ by at most $k$. We give a negative answer to a natural variant of this problem, showing that for every natural number $k$ there exists a finite set of points in the plane together with a pseudoline arrangement such that each pseudoline contains at least two points and there is a pseudoline through any pair of points where the number of points on either side of each pseudoline differ by at least $k$. Moreover, we may find such a configuration with at most $2^{2^{ck}}$ points, which, by a result of Pinchasi, is best possible up to the value of the constant $c$.

math.CO↗

arXiv2025

Even cycles in graphs avoiding longer even cycles

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 was verified by Kühn and Osthus. We show that $C_6$ and $C_{2k}$ satisfy the conjecture for all odd $k$, but observe that a recent construction of a dense $C_{10}$-free subgraph of the hypercube yields a counterexample to the conjecture for $C_8$ and $C_{10}$.

math.CO↗

arXiv2024

On the extremal number of incidence graphs

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 generalised face-incidence graphs, a family which includes the standard face-incidence graphs of regular polytopes. This builds on and generalises work of Janzer and Sudakov, who obtained the same bound for hypercubes and bipartite Kneser graphs, and allows us to confirm a conjecture of Conlon and Lee on the extremal number of $K_{r,r}$-free bipartite graphs for certain incidence graphs. In their work, Janzer and Sudakov showed that such an upper bound on the extremal number holds whenever the graph $H$ satisfies a certain percolation property which captures an appropriate sequence of repeated applications of the Cauchy--Schwarz inequality, a property which they then verify for hypercubes and bipartite Kneser graphs. This percolation property bears close resemblance to a property that arose in earlier work of Conlon and Lee on weakly norming graphs. In this latter work, Conlon and Lee developed a method for controlling repeated applications of the Cauchy--Schwarz inequality based on the properties of reflection groups, which then allowed them to isolate a broad family of weakly norming graphs. Here, we develop this method further, casting it in a purely algebraic form that allows us not only to combine it with the Janzer--Sudakov result and obtain the desired result about the extremal number of incidence graphs, but also to simplify the proofs of both the Conlon--Lee result on weakly norming graphs and a related result of Coregliano.

math.CO↗

arXiv2024

On the clique number of random Cayley graphs and related topics

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 the longstanding upper bound of $O(\log^2 N)$ due to Alon. Our proof does not make use of the underlying group structure and is purely combinatorial, with the key result being an essentially best possible upper bound for the number of subsets of given order that contain at most a given number of colors in a properly edge-colored complete graph. As a further application of this result, we study a conjecture of Alon stating that every group of order $N$ has a Cayley graph whose clique number and independence number are both $O(\log N)$, proving the conjecture for all abelian groups of order $N$ for almost all $N$. For finite vector spaces of order $N$ with characteristic congruent to $1 \pmod 4$, we prove the existence of a self-complementary Cayley graph on the vector space whose clique number and independence number are both at most $(2+o(1))\log N$. This matches the lower bound for Ramsey numbers coming from random graphs and solves, in a strong form, a problem of Alon and Orlitsky motivated by information theory.

math.CO↗

arXiv2024

On norming systems of linear equations

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 $\mathbb{F}_q^n$ for every $n>0$. For example, Gowers uniformity norms arise in this way. In this paper, we initiate the systematic study of norming linear systems by proving a range of necessary and sufficient conditions for a system to be norming. Some highlights include an isomorphism theorem for the functional $t_L(\cdot)$, a proof that any norming system must be variable-transitive and the classification of all norming systems of rank at most two.

math.CO↗

arXiv2024

Sums of linear transformations

We show that if $\mathcal{L}_1$ and $\mathcal{L}_2$ are linear transformations from $\mathbb{Z}^d$ to $\mathbb{Z}^d$ satisfying certain mild conditions, then, for any finite subset $A$ of $\mathbb{Z}^d$, $$|\mathcal{L}_1 A+\mathcal{L}_2 A|\geq \left(|\det(\mathcal{L}_1)|^{1/d}+|\det(\mathcal{L}_2)|^{1/d}\right)^d|A|- o(|A|).$$ This result corrects and confirms the two-summand case of a conjecture of Bukh and is best possible up to the lower-order term for certain choices of $\mathcal{L}_1$ and $\mathcal{L}_2$. As an application, we prove a lower bound for $|A + λ\cdot A|$ when $A$ is a finite set of real numbers and $λ$ is an algebraic number. In particular, when $λ$ is of the form $(p/q)^{1/d}$ for some $p, q, d \in \mathbb{N}$, each taken as small as possible for such a representation, we show that $$|A + λ\cdot A| \geq (p^{1/d} + q^{1/d})^d |A| - o(|A|).$$ This is again best possible up to the lower-order term and extends a recent result of Krachun and Petrov which treated the case $λ= \sqrt{2}$.

math.CO↗

arXiv2024

Sums of dilates over groups of prime order

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 $|A + λ\cdot A|$ for given $λ$, $A$ of given density $α$, and $p$ tending to infinity. We investigate this problem for $α$ fixed and $λ$ tending to infinity, proving near-optimal bounds in this case.

math.CO↗