Jacob Fox
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View article: Immersions and Albertson's conjecture
Immersions and Albertson's conjecture Open
A graph is said to contain $K_k$ (a clique of size $k$) as a weak immersion if it has $k$ vertices, pairwise connected by edge-disjoint paths. In 1989, Lescure and Meyniel made the following conjecture related to Hadwiger's conjecture: Eve…
View article: A Note on Directed Analogues of the Sidorenko and Forcing Conjectures
A Note on Directed Analogues of the Sidorenko and Forcing Conjectures Open
We study analogues of Sidorenko’s conjecture and the forcing conjecture in oriented graphs, showing that natural variants of these conjectures in directed graphs are equivalent to the asymmetric, undirected analogues of the conjectures.
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: Communities & Crowds: a toolkit for hybrid volunteering with cultural heritage collections
Communities & Crowds: a toolkit for hybrid volunteering with cultural heritage collections Open
This toolkit outlines the steps taken by the Communities & Crowds project to give ownership to local volunteers to identify photographs that reflect their interests; to digitise them to museum standards; and to share these collections with…
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: Enumeration of intersection graphs of $x$-monotone curves
Enumeration of intersection graphs of $x$-monotone curves Open
A curve in the plane is $x$-monotone if every vertical line intersects it at most once. A family of curves are called pseudo-segments if every pair of them have at most one point in common. We construct $2^{Ω(n^{4/3})}$ families, each cons…
View article: The largest subgraph without a forbidden induced subgraph
The largest subgraph without a forbidden induced subgraph Open
We initiate the systematic study of the following Turán-type question. Suppose $Γ$ is a graph with $n$ vertices such that the edge density between any pair of subsets of vertices of size at least $t$ is at most $1 - c$, for some $t$ and $c…
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: 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: Equivalence between Erdős-Hajnal and polynomial Rödl and Nikiforov conjectures
Equivalence between Erdős-Hajnal and polynomial Rödl and Nikiforov conjectures Open
It is well-known that polynomial versions of theorems of Rödl and Nikiforov, as conjectured by Fox and Sudakov and Nguyen, Scott and Seymour imply the classical Erdős-Hajnal conjecture. In this note, we prove that these three conjectures a…
View article: Variations on Sidorenko's conjecture in tournaments
Variations on Sidorenko's conjecture in tournaments Open
We study variants of Sidorenko's conjecture in tournaments, where new phenomena arise that do not have clear analogues in the setting of undirected graphs. We first consider oriented graphs that are systematically under-represented in tour…
View article: A multipartite analogue of Dilworth's Theorem
A multipartite analogue of Dilworth's Theorem Open
We prove that every partially ordered set on $n$ elements contains $k$ subsets $A_{1},A_{2},\dots,A_{k}$ such that either each of these subsets has size $Ω(n/k^{5})$ and, for every $i_{\ell}a_{2}>_{\ell}\dots>_{\ell}a_{k}$ for any $(a_1,a_…
View article: A Structure Theorem for Pseudo-Segments and Its Applications
A Structure Theorem for Pseudo-Segments and Its Applications Open
We prove a far-reaching strengthening of Szemerédi’s regularity lemma for intersection graphs of pseudo-segments. It shows that the vertex set of such graphs can be partitioned into a bounded number of parts of roughly the same size such t…
View article: Ramsey and Turán numbers of sparse hypergraphs
Ramsey and Turán numbers of sparse hypergraphs Open
Degeneracy plays an important role in understanding Turán- and Ramsey-type properties of graphs. Unfortunately, the usual hypergraphical generalization of degeneracy fails to capture these properties. We define the skeletal degeneracy of a…
View article: The growth rate of multicolor Ramsey numbers of $3$-graphs
The growth rate of multicolor Ramsey numbers of $3$-graphs Open
The $q$-color Ramsey number of a $k$-uniform hypergraph $G,$ denoted $r(G;q)$, is the minimum integer $N$ such that any coloring of the edges of the complete $k$-uniform hypergraph on $N$ vertices contains a monochromatic copy of $G$. The …
View article: Triangle Ramsey numbers of complete graphs
Triangle Ramsey numbers of complete graphs Open
A graph is $H$-Ramsey if every two-coloring of its edges contains a monochromatic copy of $H$. Define the $F$-Ramsey number of $H$, denoted by $r_F(H)$, to be the minimum number of copies of $F$ in a graph which is $H$-Ramsey. This general…
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: Extremal results on feedback arc sets in digraphs
Extremal results on feedback arc sets in digraphs Open
For an oriented graph , let denote the size of a minimum feedback arc set , a smallest edge subset whose deletion leaves an acyclic subgraph. Berger and Shor proved that any ‐edge oriented graph satisfies . We observe that if an oriented g…
View article: Ramsey numbers of hypergraphs of a given size
Ramsey numbers of hypergraphs of a given size Open
The $q$-color Ramsey number of a $k$-uniform hypergraph $H$ is the minimum integer $N$ such that any $q$-coloring of the complete $k$-uniform hypergraph on $N$ vertices contains a monochromatic copy of $H$. The study of these numbers is on…
View article: Ramsey goodness of books revisited
Ramsey goodness of books revisited Open
The Ramsey number $r(G,H)$ is the minimum $N$ such that every graph on $N$ vertices contains $G$ as a subgraph or its complement contains $H$ as a subgraph. For integers $n \geq k \geq 1$, the $k$-book $B_{k,n}$ is the graph on $n$ vertice…
View article: Ramsey multiplicity and the Turán coloring
Ramsey multiplicity and the Turán coloring Open
Extending an earlier conjecture of Erd\H{o}s, Burr and Rosta conjectured that among all two-colorings of the edges of a complete graph, the uniformly random coloring asymptotically minimizes the number of monochromatic copies of any fixed …
View article: Induced subgraph density. II. Sparse and dense sets in cographs
Induced subgraph density. II. Sparse and dense sets in cographs Open
A well-known theorem of Rödl says that for every graph $H$, and every $ε>0$, there exists $δ>0$ such that if $G$ does not contain an induced copy of $H$, then there exists $X\subseteq V(G)$ with $|X|\ge δ|G|$ such that one of $G[X],\overli…
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: 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: Quasiplanar Graphs, String Graphs, and the Erdős-Gallai Problem
Quasiplanar Graphs, String Graphs, and the Erdős-Gallai Problem Open
An $r$-quasiplanar graph is a graph drawn in the plane with no $r$ pairwise crossing edges. Let $s \geq 3$ be an integer and $r=2^s$. We prove that there is a constant $C$ such that every $r$-quasiplanar graph with $n \geq r$ vertices has …