The Schur degree of additive sets

TL;DR

This paper introduces the Schur degree as a new measure for subsets of abelian groups, extending the classical sumfree concept, and explores its implications for bounding Schur numbers with conjectured inequalities.

Contribution

It generalizes the concept of sumfree sets to Schur degree, establishes a broader inequality involving Schur numbers and Ramsey numbers, and proposes a conjecture to improve bounds on Schur numbers.

Findings

The classical inequality S(n) ≤ R(n,3) - 2 is valid in a broader context.

A new conjecture S(n) ≤ n(S(n-1)+1) is proposed to improve bounds on Schur numbers.

Current bounds for S(6) are significantly widened, with potential for tighter estimates if the conjecture holds.

Abstract

Let (G, +) be an abelian group. A subset of G is sumfree if it contains no elements x, y, z such that x +y = z. We extend this concept by introducing the Schur degree of a subset of G, where Schur degree 1 corresponds to sumfree. The classical inequality S(n) $\le$ R n (3) -- 2, between the Schur number S(n) and the Ramsey number R n (3) = R(3,. .. , 3), is shown to remain valid in a wider context, involving the Schur degree of certain subsets of G. Recursive upper bounds are known for R n (3) but not for S(n) so far. We formulate a conjecture which, if true, would fill this gap. Indeed, our study of the Schur degree leads us to conjecture S(n) $\le$ n(S(n -- 1) + 1) for all n $\ge$ 2. If true, it would yield substantially better upper bounds on the Schur numbers, e.g. S(6) $\le$ 966 conjecturally, whereas all is known so far is 536 $\le$ S(6) $\le$ 1836.