# On the independence number of $(3, 3)$-Ramsey graphs and the Folkman number $F_e(3, 3; 4)$

## TL;DR

This paper establishes an upper bound on the independence number of certain Ramsey graphs and uses this to improve the lower bound of the Folkman number F_e(3, 3; 4) from 20 to 21 through theoretical and computational methods.

## Contribution

It proves a new upper bound on the independence number of (3,3)-Ramsey graphs without 4-cliques and improves the lower bound of the Folkman number F_e(3, 3; 4).

## Key findings

- Proved that for such graphs, lpha(G)  n - 16.
- Established that F_e(3, 3; 4)  21.
- Used computational methods to support the new bound.

## Abstract

The graph $G$ is called a $(3, 3)$-Ramsey graph if in every coloring of the edges of $G$ in two colors there is a monochromatic triangle. The minimum number of vertices of the $(3, 3)$-Ramsey graphs without 4-cliques is denoted by $F_e(3, 3; 4)$. The number $F_e(3, 3; 4)$ is referred to as the most wanted Folkman number. It is known that $20 \leq F_e(3, 3; 4) \leq 786$. In this paper we prove that if $G$ is an $n$-vertex $(3, 3)$-Ramsey graph without 4-cliques, then $\alpha(G) \leq n - 16$, where $\alpha(G)$ denotes the independence number of $G$. Using the newly obtained bound on $\alpha(G)$ and complex computer calculations we obtain the new lower bound $$F_e(3, 3; 4) \geq 21.$$