# Semidefinite Programming and Ramsey Numbers

## TL;DR

This paper develops a novel technique using flag algebra to determine exact small Ramsey numbers, successfully establishing new bounds and exact values for several specific cases, challenging the notion that the method is only for large graphs.

## Contribution

The paper introduces a new approach with flag algebras to find exact small Ramsey numbers, expanding the method's applicability beyond large graphs.

## Key findings

- Established exact values for several small Ramsey numbers.
- Developed a technique to adapt flag algebra for small graph problems.
- Provided new upper bounds for multiple small graph and hypergraph Ramsey numbers.

## Abstract

Finding exact Ramsey numbers is a problem typically restricted to relatively small graphs. The flag algebra method was developed to find asymptotic results for very large graphs, so it seems that the method is not suitable for finding small Ramsey numbers. But this intuition is wrong, and we will develop a technique to do just that in this paper. We find new upper bounds for many small graph and hypergraph Ramsey numbers. As a result, we prove the exact values $R(K_4^-,K_4^-,K_4^-)=28$, $R(K_8,C_5)= 29$, $R(K_9,C_6)= 41$, $R(Q_3,Q_3)=13$, $R(K_{3,5},K_{1,6})=17$, $R(C_3, C_5, C_5)= 17$, and $R(K_4^-,K_5^-;3)= 12$. We hope that this technique will be adapted to address other questions for smaller graphs with the flag algebra method.