Tighter Bounds on Directed Ramsey Number R(7)

TL;DR

This paper improves bounds on the directed Ramsey number R(7) by classifying TT_6-free tournaments and using computer-assisted proofs and SAT technology to refine the minimum tournament size guaranteeing a transitive subtournament of size 7.

Contribution

It provides the first computer-assisted proof of a conjecture on TT_6-free tournaments and establishes tighter bounds on R(7) using classification and SAT methods.

Findings

All TT_6-free tournaments on 23 vertices classified.

TT_6-free tournaments on 24 and 25 vertices are subgraphs of ST_{27}.

Improved bounds: 34 ≤ R(7) ≤ 47.

Abstract

Tournaments are orientations of the complete graph, and the directed Ramsey number $R(k)$ is the minimum number of vertices a tournament must have to be guaranteed to contain a transitive subtournament of size $k$, which we denote by $TT_k$. We include a computer-assisted proof of a conjecture by Sanchez-Flores that all $TT_6$-free tournaments on 24 and 25 vertices are subtournaments of $ST_{27}$, the unique largest TT_6-free tournament. We also classify all $TT_6$-free tournaments on 23 vertices. We use these results, combined with assistance from SAT technology, to obtain the following improved bounds: $34 \leq R(7) \leq 47$.