# The Ramsey Number of Fano Plane Versus Tight Path

## TL;DR

This paper investigates the hypergraph Ramsey number involving the Fano plane and shows that the tight path of length n is -good, advancing understanding of hypergraph colorings and Ramsey numbers.

## Contribution

It proves that the tight path of length n is -good, providing new insights into the structure of hypergraph Ramsey numbers involving the Fano plane.

## Key findings

- The tight path of length n is -good.
- Progress on Conlon's question about -good hypergraphs.
- Enhanced understanding of hypergraph Ramsey numbers involving the Fano plane.

## Abstract

The hypergraph Ramsey number of two $3$-uniform hypergraphs $G$ and $H$, denoted by $R(G,H)$, is the least integer $N$ such that every red-blue edge-coloring of the complete $3$-uniform hypergraph on $N$ vertices contains a red copy of $G$ or a blue copy of $H$. The Fano plane $\mathbb{F}$ is the unique 3-uniform hypergraph with seven edges on seven vertices in which every pair of vertices is contained in a unique edge. There is a simple construction showing that $R(H,\mathbb{F}) \ge 2(v(H)-1) + 1.$ Hypergraphs $H$ for which the equality holds are called $\mathbb{F}$-good. Conlon asked to determine all $H$ that are $\mathbb{F}$-good. In this short paper we make progress on this problem and prove that the tight path of length $n$ is $\mathbb{F}$-good.