arXiv subjects

Jacques Verstraëte

Publications and source records attributed to Jacques Verstraëte.

At least 19 recordsLinked to original sources

arXiv2026

Spanning subhypergraphs with degree constraints

An old result of Tutte states that any $d$-regular graph contains a spanning subgraph in which every vertex has degree $k$ or $k+1$, for every $1\leq k\leq d$. We generalize this statement to hypergraphs, showing, for example, that every $3$-uniform $d$-regular hypergraph contains a subgraph in which all degrees are $k, k+1$ or $k+2$, for every $1\leq k\leq d$. This statement is best possible in the sense that the corresponding statement with only two allowed consecutive values is not true. We provide generalizations of this statement to higher uniformities and discuss several open problems.

math.CO↗

arXiv2026

Counting Hypergraphs with Large Girth

Morris and Saxton used the method of containers to bound the number of $n$-vertex graphs with $m$ edges containing no $\ell$-cycles, and hence graphs of girth more than $\ell$. We consider a generalization to $r$-uniform hypergraphs. The {\em girth} of a hypergraph $H$ is the minimum $\ell$ such that for some $F \subseteq H$, there exists a bijection $ϕ: E(C_\ell) \to E(F)$ with $e\subseteq ϕ(e)$ for all $e\in E(C_\ell)$. Letting $N_m^r(n,\ell)$ denote the number of $n$-vertex $r$-uniform hypergraphs with $m$ edges and girth larger than $\ell$ and defining $λ= \lceil (r - 2)/(\ell - 2)\rfloor$, we show \[ N_m^r(n,\ell) \leq N_m^2(n,\ell)^{r - 1 + λ}\] which is tight when $\ell - 2 $ divides $r - 2$ up to a $1 + o(1)$ term in the exponent. This result is used to address the extremal problem for subgraphs of girth more than $\ell$ in random $r$-uniform hypergraphs.

math.CO↗

arXiv2026

On the Erdős-Rogers function

We show that the Erdős-Rogers function $f_{s,s+1}(n)$ satisfies $$f_{s,s+1}(n) = Θ( \sqrt{n \log n} )$$ for every $s \ge 2$. More precisely, we construct a $K_{s+1}$-free graph on $n$ vertices in which every set of at least $C(s)\sqrt{n \log n}$ vertices contains a copy of $K_s$ for some constant $C(s)$, which implies the upper bound. The matching lower bound follows from a theorem of Joret, Micek, Reed and Smid on the clique chromatic number of a graph.

math.CO↗

arXiv2026

Hypergraph Ramsey numbers with quasipolynomial growth rate

For a 3-uniform hypergraph (3-graph) $F$, let $r(F,n)$ be the smallest $N$ such that any $N$-vertex $F$-free 3-graph has an independent set of size $n$. We construct a $3$-graph $H_2$ with six vertices and five edges such that $r(H_2,n)=n^{Θ(\log n)}$, and a more general family of $3$-graphs $F$ for which $r(F,n)=n^{\log^{Θ(1)}(n)}$. These are the first examples of such Ramsey number known to be neither polynomial nor exponential.

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↗

arXiv2024

Off-diagonal Ramsey numbers for slowly growing hypergraphs

For a $k$-uniform hypergraph $F$ and a positive integer $n$, the Ramsey number $r(F,n)$ denotes the minimum $N$ such that every $N$-vertex $F$-free $k$-uniform hypergraph contains an independent set of $n$ vertices. A hypergraph is $\textit{slowly growing}$ if there is an ordering $e_1,e_2,\dots,e_t$ of its edges such that $|e_i \setminus \bigcup_{j = 1}^{i - 1}e_j| \leq 1$ for each $i \in \{2, \ldots, t\}$. We prove that if $k \geq 3$ is fixed and $F$ is any non $k$-partite slowly growing $k$-uniform hypergraph, then for $n\ge2$, \[ r(F,n) = Ω\Bigl(\frac{n^k}{(\log n)^{2k - 2}}\Bigr).\] In particular, we deduce that the off-diagonal Ramsey number $r(F_5,n)$ is of order $n^{3}/\mbox{polylog}(n)$, where $F_5$ is the triple system $\{123, 124, 345\}$. This is the only 3-uniform Berge triangle for which the polynomial power of its off-diagonal Ramsey number was not previously known. Our constructions use pseudorandom graphs, martingales, and hypergraph containers.

math.CO↗

arXiv2024

Ramsey numbers and the Zarankiewicz problem

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 $m$ by $n$ $0/1$-matrix that does not have any matrix from a fixed finite family $\mathcal{L}(F)$ derived from $F$ as a submatrix. As an application, we give new lower bounds for the Ramsey numbers $r(C_5,t)$ and $r(C_7,t)$, namely, $r(C_5,t) = \tildeΩ(t^{\frac{10}{7}})$ and $r(C_7,t) = \tildeΩ(t^{\frac{5}{4}})$. We also show how the truth of a plausible conjecture about Zarankiewicz numbers would allow an approximate determination of $r(C_{2\ell+1}, t)$ for any fixed integer $\ell \geq 2$.

math.CO↗

arXiv2022

Large $(k; r, s; n, q)$-sets in Projective Spaces

A $(k; r, s; n, q)$-set (short: $(r,s)$-set) of $\mathrm{PG}(n, q)$ is a set of points $X$ with $|X| = k$ such that no $s$-space contains more than $r$ points of $X$. We investigate the asymptotic size of $(r, s)$-sets for $n$ fixed and $q \rightarrow \infty$. In particular, we show the existence of $(3, 2)$-sets of size $(1+o(1)) q^{3/2}$ for $n=6$, $(4, 2)$-sets of size $(1+o(1)) q^{\frac{n-1}{2}}$, and $(9, 2)$-sets of size $(1+o(1)) q^2$ for $n=4$. We also generalize a bound by Rao from 1947 and show that an $(r,s)$-set has size at most $O(q^{\frac{n-e+1}{e}})$ if there exist integers $d,e \geq 2$ such that $s=d(e-1)$ and $r=de-1$.

math.CO↗

arXiv2021

On Asymptotic Packing of Geometric Graphs

A set of geometric graphs is {\em geometric-packable} if it can be asymptotically packed into every sequence of drawings of the complete graph $K_n$. For example, the set of geometric triangles is geometric-packable due to the existence of Steiner Triple Systems. When $G$ is the $4$-cycle (or $4$-cycle with a chord), we show that the set of plane drawings of $G$ is geometric-packable. In contrast, the analogous statement is false when $G$ is nearly any other planar Hamiltonian graph (with at most 3 possible exceptions). A convex geometric graph is {\em convex-packable} if it can be asymptotically packed into the convex drawings of the complete graphs. For each planar Hamiltonian graph $G$, we determine whether or not a plane $G$ is convex-packable. Many of our proofs explicitly construct these packings; in these cases, the packings exhibit a symmetry that mirrors the vertex transitivity of $K_n$.

math.CO↗

arXiv2021

Ramsey Numbers for Non-trivial Berge Cycles

In this paper, we consider an extension of cycle-complete graph Ramsey numbers to Berge cycles in hypergraphs: for $k \geq 2$, a {\em non-trivial Berge $k$-cycle} is a family of sets $e_1,e_2,\dots,e_k$ such that $e_1 \cap e_2, e_2 \cap e_3,\dots,e_k \cap e_1$ has a system of distinct representatives and $e_1 \cap e_2 \cap \dots \cap e_k = \emptyset$. In the case that all the sets $e_i$ have size three, let $\mathcal{B}_k$ denotes the family of all non-trivial Berge $k$-cycles. The {\em Ramsey numbers} $R(t,\mathcal{B}_k)$ denote the minimum $n$ such that every $n$-vertex $3$-uniform hypergraph contains either a non-trivial Berge $k$-cycle or an independent set of size $t$. We prove \[ R(t, \mathcal{B}_{2k}) \leq t^{1 + \frac{1}{2k-1} + \frac{4}{\sqrt{\log t}}}\] and moreover, we show that if a conjecture of Erdős and Simonovits \cite{ES} on girth in graphs is true, then this is tight up to a factor $t^{o(1)}$ as $t \rightarrow \infty$.

math.CO↗

arXiv2021

Relative Turán Problems for Uniform Hypergraphs

For two graphs $F$ and $H$, the relative Turán number $\mathrm{ex}(H,F)$ is the maximum number of edges in an $F$-free subgraph of $H$. Foucaud, Krivelevich, and Perarnau \cite{FKP} and Perarnau and Reed \cite{PR} studied these quantities as a function of the maximum degree of $H$. In this paper, we study a generalization for uniform hypergraphs. If $F$ is a complete $r$-partite $r$-uniform hypergraph with parts of sizes $s_1,s_2,\dots,s_r$ with each $s_{i + 1}$ sufficiently large relative to $s_i$, then with $1/β= \sum_{i = 2}^r \prod_{j = 1}^{i - 1} s_j$ we prove that for any $r$-uniform hypergraph $H$ with maximum degree $Δ$, \[\mathrm{ex}(H,F)\ge Δ^{-β- o(1)} \cdot e(H).\] This is tight as $Δ\rightarrow \infty$ up to the $o(1)$ term in the exponent, since we show there exists a $Δ$-regular $r$-graph $H$ such that $\mathrm{ex}(H,F)=O(Δ^{-β}) \cdot e(H)$. Similar tight results are obtained when $H$ is the random $n$-vertex $r$-graph $H_{n,p}^r$ with edge-probability $p$, extending results of Balogh and Samotij \cite{BS} and Morris and Saxton \cite{MS}.

math.CO↗

arXiv2020

Extremal problems for pairs of triangles

A convex geometric hypergraph or cgh consists of a family of subsets of a strictly convex set of points in the plane. There are eight pairwise nonisomorphic cgh's consisting of two disjoint triples. These were studied at length by Braß (2004) and by Aronov, Dujmović, Morin, Ooms, and da Silveira (2019). We determine the extremal functions exactly for seven of the eight configurations. The above results are about cyclically ordered hypergraphs. We extend some of them for triangle systems with vertices from a non-convex set. We also solve problems posed by P. Frankl, Holmsen and Kupavskii (2020), in particular, we determine the exact maximum size of an intersecting family of triangles whose vertices come from a set of $n$ points in the plane.

math.CO↗

arXiv2020

A note on $k$-wise oddtown problems

For integers $2 \leq t \leq k$, we consider a collection of $k$ set families $\mathcal{A}_j: 1 \leq j \leq k$ where $\mathcal{A}_j = \{ A_{j,i} \subseteq [n] : 1 \leq i \leq m \}$ and $|A_{1, i_1} \cap \cdots \cap A_{k,i_k}|$ is even if and only if at least $t$ of the $i_j$ are distinct. In this paper, we prove that $m =O(n^{ 1/ \lfloor k/2 \rfloor})$ when $t=k$ and $m = O( n^{1/(t-1)})$ when $2t-2 \leq k$ and prove that both of these bounds are best possible. Specializing to the case where $\mathcal{A} = \mathcal{A}_1 = \cdots = \mathcal{A}_k$, we recover a variation of the classical oddtown problem.

math.CO↗

arXiv2020

Partitioning ordered hypergraphs

An {\em ordered $r$-graph} is an $r$-uniform hypergraph whose vertex set is linearly ordered. Given $2\leq k\leq r$, an ordered $r$-graph $H$ is {\em interval} $k$-{\em partite} if there exist at least $k$ disjoint intervals in the ordering such that every edge of $H$ has nonempty intersection with each of the intervals and is contained in their union. Our main result implies that for each $α> k - 1$ and $d>0$, every $n$-vertex ordered $r$-graph with $d \,n^α$ edges has for some $m\leq n$ an $m$-vertex interval $k$-partite subgraph with $Ω(d\, m^α)$ edges. This is an extension to ordered $r$-graphs of the observation by Erd\H os and Kleitman that every $r$-graph contains an $r$-partite subgraph with a constant proportion of the edges. The restriction $α> k-1$ is sharp. We also present applications of the main result to several extremal problems for ordered hypergraphs.

math.CO↗

arXiv2020

Tight paths in convex geometric hypergraphs

In this paper, we prove a theorem on tight paths in convex geometric hypergraphs, which is asymptotically sharp in infinitely many cases. Our geometric theorem is a common generalization of early results of Hopf and Pannwitz [12], Sutherland [19], Kupitz and Perles [16] for convex geometric graphs, as well as the classical Erdős-Gallai Theorem [6] for graphs. As a consequence, we obtain the first substantial improvement on the Turán problem for tight paths in uniform hypergraphs.

math.CO↗

arXiv2020

Extremal problems for hypergraph blowups of trees

In this paper we present a novel approach in extremal set theory which may be viewed as an asymmetric version of Katona's permutation method. We use it to find more Turán numbers of hypergraphs in the Erdős--Ko--Rado range. An $(a,b)$-path $P$ of length $2k-1$ consists of $2k-1$ sets of size $r=a+b$ as follows. Take $k$ pairwise disjoint $a$-element sets $A_0, A_2, \dots, A_{2k-2}$ and other $k$ pairwise disjoint $b$-element sets $B_1, B_3, \dots, B_{2k-1}$ and order them linearly as $A_0, B_1, A_2, B_3, A_4\dots$. Define the (hyper)edges of $P_{2k-1}(a,b)$ as the sets of the form $A_i\cup B_{i+1}$ and $B_j\cup A_{j+1}$. The members of $P$ can be represented as $r$-element intervals of the $ak+bk$ element underlying set. Our main result is about hypergraphs that are blowups of trees, and implies that for fixed $k,a,b$, as $n\to \infty$ \[ {\rm ex}_r(n,P_{2k-1}(a,b)) = (k - 1){n \choose r - 1} + o(n^{r - 1}).\] This generalizes the Erdős--Gallai theorem for graphs which is the case of $a=b=1$. We also determine the asymptotics when $a+b$ is even; the remaining cases are still open.

math.CO↗

arXiv2019

Constructions of point-line arrangements in the plane with large girth

A classical result by Erdős, and later on by Bondy and Simonivits, states that every $n$-vertex graph with no cycle of length $2k$ has at most $O(n^{1+1 /k})$ edges. This bound is known to be tight when $k \in \{2,3,5\},$ but it is a major open problem in extremal graph theory to decide if this bound is tight for all $k$. In this paper, we study the effect of forbidding short even cycles in incidence graphs of point-line arrangements in the plane. It is not known if the Erdős upper bound stated above can be improved to $o(n^{1+1/k})$ in this geometric setting, and in this note, we establish non-trivial lower bounds for this problem by modifying known constructions arising in finite geometries. In particular, by modifying a construction due to Labeznik and Ustimenko, we construct an arrangement of $n$ points and $n$ lines in the plane, such that their incidence graph has girth at least $k + 5$, and determines at least $Ω({n^{1+\frac{4}{k^2+6k-3}}})$ incidences. We also apply the same technique to Wenger graphs, which gives a better lower bound for $k=5.$

math.CO↗