Schur’s Problem of sum-free sets emerged in the early 1900’s in an attempt to prove Fermat’s Last Theorem

Schur’s Problem of Sum-free Sets and the Modified Abbott Conjecture

By Andrew Matteson

Schur’s Problem of sum-free sets emerged in the early 1900’s in an attempt to prove Fermat’s Last Theorem. Schur was not able to do this; however, his ideas and the proofs surrounding them formed a precursor to Ramsey Theory. Schur’s problem still plays a significant role in multicolor Ramsey Theory.

This talk will focus on the history and future of Schur’s problem, two variations of the original problem, conjectures of several mathematicians about the relationship of the variations, and the importance of Schur’s problem with respect to several subjects within combinatorics. A brief description of subject matter is as follows.

A set, A_r is said to be sum free if for any two elements a and b of A_r a+b is not contained in A_r. Let f be a function mapping the integer interval [1, n] into a finite number of sets A₁…A_i. Schur proved that for any number i, there is a finite number S(n) such that no such f exists for the integer interval [1,S(n)+1]. Schur provided some general bounds on S(n) and calculated S(1), S(2) and S(3). S(4) was found in the 1965 by Baumert.

In the late 1920’s following the work of Schur, Frank Plumpton Ramsey began a search for the guaranteed degree of order in large random systems. In the context of his work and the continuation of it made by Paul Erdos, the Schur numbers form bounds on another special set of numbers R_n(3). This relationship allowed Erdos and more recently Fredrickson, Sweet and Exoo to establish lower bounds on several of these important numbers.

The Abbott conjecture is based off of a modular variation of Schur’s problem. It states that at least one sum-free partition of [1, S(n)] is sum-free modulo n+1. At first glance this seems to be impossible; however, there is quite a bit of evidence to the contrary. We offer a proof of the Abbott Conjecture dependent on certain assumptions about symmetry of sum-free partitions that empirical evidence supports. We discuss proposed techniques on how to prove these assumptions.

We will also discuss a proof of a modified Abbot Conjecture that has implications on complexity theory and algorithm efficiency. Time permitting we will address improved computational techniques for S(n) that employ hybrids of artificial intelligence and backtrack heuristics, developed as a result of this proof.

The project in its present form is a continuation of research begun under the mentorship of Dr. Eugene Curtin of Texas State University during the 2003 Mathworks summer program. The presenter would like to thank Dr. Curtin and Dr. Warshauer for their continued support and encouragement.