401 and beyond: improved bounds and algorithms for the Ramsey algebra search

TL;DR

This paper improves algorithms for finding Ramsey algebras over finite fields and establishes bounds on the prime moduli, demonstrating the non-existence of certain partitions for prime p.

Contribution

It introduces enhanced algorithms for searching Ramsey algebras and proves an upper bound on prime moduli related to coset partitions.

Findings

No prime p allows a 13-color Ramsey algebra with 13 cosets.

Derived an upper bound on p based on the number of cosets.

Improved search bounds for Ramsey algebra constructions.

Abstract

In this paper, we discuss an improvement of an algorithm to search for primes $p$ and coset-partitions of Z/pZ* that yield Ramsey algebras over Z/pZ. We also prove an upper bound on the modulus p in terms of the number of cosets. We have, as a corollary, that there is no prime $p$ for which there exists a partition of Z/pZ* into 13 cosets that yields a 13-color Ramsey algebra. Thus A263308(13) = 0.