# On Ramsey numbers of hedgehogs

## TL;DR

This paper proves that the two-color Ramsey number of the hedgehog hypergraph grows nearly linearly with the number of vertices, specifically as O(t^2 log t), answering a question about its growth rate.

## Contribution

The authors establish an upper bound of O(t^2 log t) for the two-color Ramsey number of hedgehogs, improving understanding of its growth and confirming near-linear behavior.

## Key findings

- Two-color Ramsey number of hedgehogs is O(t^2 log t)
- Answers affirmatively to the near-linear growth question
- Provides bounds contrasting with exponential four-color Ramsey numbers

## Abstract

The hedgehog $H_t$ is a 3-uniform hypergraph on vertices $1,\dots,t+\binom{t}{2}$ such that, for any pair $(i,j)$ with $1\le i<j\le t$, there exists a unique vertex $k>t$ such that $\{i,j,k\}$ is an edge. Conlon, Fox, and R\"odl proved that the two-color Ramsey number of the hedgehog grows polynomially in the number of its vertices, while the four-color Ramsey number grows exponentially in the number of its vertices. They asked whether the two-color Ramsey number of the hedgehog $H_t$ is nearly linear in the number of its vertices. We answer this question affirmatively, proving that $r(H_t) = O(t^2\ln t)$.