About

Jacob Fox is a professor in the Department of Mathematics at Stanford University. His research areas include combinatorics and probability. He is involved in advising students such as Carl Schildkraut and Shengtong Zhang. For further contact, he can be reached via email at jacobfox@stanford.edu or by phone at (650) 736-6988. His office is located in Building 380, 383-K at Stanford University.

Research topics

• Combinatorics
• Mathematics
• Discrete mathematics
• Computer science
• Arithmetic

Selected publications

• Separators for intersection graphs of spheres

  ArXiv.org · 2026-03-23

  articleOpen access1st authorCorresponding

  We prove the existence of optimal separators for intersection graphs of balls and spheres in any dimension $d$. One of our results is that if an intersection graph of $n$ spheres in $\mathbb{R}^d$ has $m$ edges, then it contains a balanced separator of size $O_d(m^{1/d}n^{1-2/d})$. This bound is best possible in terms of the parameters involved. The same result holds if the balls and spheres are replaced by fat convex bodies and their boundaries.

  PublisherOA PDF

• Finer control on relative sizes of iterated sumsets

  ArXiv.org · 2025-06-06

  preprintOpen access1st authorCorresponding

  Inspired by recent questions of Nathanson, we show that for any infinite abelian group $G$ and any integers $m_1, \ldots, m_H$, there exist finite subsets $A,B \subseteq G$ such that $|hA|-|hB|=m_h$ for each $1 \leq h \leq H$. We also raise, and begin to address, questions about the smallest possible cardinalities and diameters of such sets $A,B$.

  PublisherOA PDFDOI

• On Off-Diagonal Hypergraph Ramsey Numbers

  International Mathematics Research Notices · 2025-05-23 · 2 citations

  article

  Abstract 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 complete $3$-uniform hypergraph with $n$ vertices. We study this problem, proving two main results. First, we show that for a broad class of $H$, including links of odd cycles and tight cycles of length not divisible by three, $r(H, K_{n}^{(3)}) \ge 2^{\Omega _{H}(n \log n)}$. This significantly generalizes and simplifies an earlier construction of Fox and He which handled the case of links of odd cycles and is sharp both in this case and for all but finitely many tight cycles of length not divisible by three. Second, disproving a folklore conjecture in the area, we show that there exists a linear hypergraph $H$ for which $r(H, K_{n}^{(3)})$ is superpolynomial in $n$. This provides the first example of a separation between $r(H,K_{n}^{(3)})$ and $r(H,K_{n,n,n}^{(3)})$, since the latter is known to be polynomial in $n$ when $H$ is linear.

  PublisherDOI

• Triangle Ramsey numbers of complete graphs

  Journal of Combinatorial Theory Series B · 2025-10-08 · 1 citations

  article1st authorCorresponding
  PublisherDOI

• Big line or big convex polygon

  Computational Geometry · 2025-08-26

  articleCorresponding
  PublisherDOI

• Color-avoiding directed paths in tournaments

  ArXiv.org · 2025-12-11

  preprintOpen access1st authorCorresponding

  We study the following Ramsey-theoretic question: given a $q$-coloring of the edges of a tournament, how long of a directed path can we guarantee whose edges avoid one of the colors? Questions of this type have applications in many areas, such as vector sequences, convex geometry, and extremal hypergraph theory, and have been extensively studied over the past 50 years. We prove that if $\varepsilon>0$ is fixed and $q$ is sufficiently large, then every $q$-edge-colored $N$-vertex tournament contains a color-avoiding directed path of length $N^{1-\varepsilon}$. This answers a question of Gowers and Long, strengthens several of their results, and extends earlier work of Loh.

  PublisherOA PDFDOI

• A structure theorem for pseudosegments and its applications

  Journal of Combinatorial Theory Series B · 2025-05-06

  articleOpen access1st authorCorresponding
  PublisherDOI

• The Largest Subgraph Without A Forbidden Induced Subgraph

  COMBINATORICA · 2025-11-07

  article1st authorCorresponding
  PublisherDOI

• Immersions and Albertson's conjecture

  ArXiv.org · 2025-10-07

  preprintOpen access1st authorCorresponding

  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: Every graph of chromatic number $k$ contains $K_k$ as a weak immersion. We prove this conjecture for graphs with at most $(1.64-o(1))k$ vertices. As an application, we make some progress on Albertson's conjecture, according to which every graph $G$ with chromatic number $k$ satisfies $cr(G) \geq cr(K_k)$. In particular, we show that the conjecture is true for all graphs of chromatic number $k$, provided that they have at most $(1.64-o(1))k$ vertices.

  PublisherOA PDFDOI

• When are off-diagonal hypergraph Ramsey numbers polynomial?

  Proceedings of the American Mathematical Society · 2025-09-19

  article

  A natural open problem in Ramsey theory is to determine those <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"> <mml:semantics> <mml:mn>3</mml:mn> <mml:annotation encoding="application/x-tex">3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -graphs <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which the off-diagonal Ramsey number <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r left-parenthesis upper H comma upper K Subscript n Superscript left-parenthesis 3 right-parenthesis Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>H</mml:mi> <mml:mo>,</mml:mo> <mml:msubsup> <mml:mi>K</mml:mi> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">r(H, K_n^{(3)})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> grows polynomially with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We make substantial progress on this question by showing that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is tightly connected or has at most two tight components, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="r left-parenthesis upper H comma upper K Subscript n Superscript left-parenthesis 3 right-parenthesis Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>H</mml:mi> <mml:mo>,</mml:mo> <mml:msubsup> <mml:mi>K</mml:mi> <mml:mi>n</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">r(H, K_n^{(3)})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> grows polynomially if and only if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is contained in an iterated blowup of an edge.

  PublisherDOI

Frequent coauthors

• David Conlon

  California Institute of Technology

  138 shared

• Benny Sudakov

  ETH Zurich

  137 shared

• János Pach

  Alfréd Rényi Institute of Mathematics

  107 shared

• Andrew Suk

  University of California, San Diego

  66 shared

• Yufei Zhao

  34 shared

• Yuval Wigderson

  23 shared

• Fan Wei

  23 shared

• Huy Tuan Pham

  Stanford University

  22 shared

Labs

• Jacob FoxPI