# The Graph Ramsey Number $R(F_\ell,K_6)$

## TL;DR

This paper proves a conjecture about the exact value of a specific graph Ramsey number involving a join of a single vertex and disjoint edges, confirming the formula for the case when n=6.

## Contribution

It confirms the conjecture for the case n=6, establishing the exact Ramsey number for the pair (F_ell, K_6).

## Key findings

- Proved the conjecture R(F_ell, K_6) = 2\,ell\, (6-1) + 1 for all \,ell \, ≥ 6.
- Established the exact value of the Ramsey number for the specific graph pair.
- Extended understanding of Ramsey numbers involving joins of graphs.

## Abstract

For a given pair of two graphs $(F,H)$, let $R(F,H)$ be the smallest positive integer $r$ such that for any graph $G$ of order $r$, either $G$ contains $F$ as a subgraph or the complement of $G$ contains $H$ as a subgraph. Baskoro, Broersma and Surahmat (2005) conjectured that \[ R(F_\ell,K_n)=2\ell(n-1)+1 \] for $\ell\ge n\ge3$, where $F_\ell$ is the join of $K_1$ and $\ell K_2$. In this paper, we prove that this conjecture is true for the case $n=6$.