Jacob Fox

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4 papers · 1 filter

math.CO2026

A new lower bound for two-color van der Waerden numbers

Marcelo Campos, Jacob Fox, Carl Schildkraut

The van der Waerden number $w (k)$ is the smallest positive integer $N$ such that every two-coloring of ${1, 2, \dots, N}$ contains a monochromatic $k$ -term arithmetic progression.…

math.CO2026

On nearly consecutive sequences without long arithmetic progressions

Jacob Fox, Carl Schildkraut

A sequence $a_{1}, \dots, a_{n}$ of integers is nearly consecutive if $a_{i + 1} - a_{i} \in {1, 2}$ for $1 \leq i \leq n - 1$ . We prove that there are nearly consecutive sequences of length $…

math.CO2024

Induced subgraph density. II. Sparse and dense sets in cographs

Jacob Fox, Tung Nguyen, Alex Scott +1

A well-known theorem of RÃ¶dl says that for every graph $H$ , and every $I^µ > 0$ , there exists $I^´ > 0$ such that if $G$ does not contain an induced copy of $H$ , then there exists $X\su…

math.CO2024

Acyclic subgraphs of tournaments with high chromatic number

Jacob Fox, Matthew Kwan, Benny Sudakov

We prove that every $n$ -vertex tournament $G$ has an acyclic subgraph with chromatic number at least $n^{5/9 - o (1)}$ , while there exists an $n$ -vertex tournament $G$ whose every acy…