A lower bound for weak Schur numbers with a deterministic algorithm

Abstract

Given a positive integer K, the weak Schur problem is to find the largest weakly sum-free K-partition of positive integers. Our goal is to increase known values of n for which the set {1,…,n} can be divided into sum-free sets, i.e. sets for which there are no three distinct elements x, y, z in the same set such that x+y=z.Our main contribution consists in designing a recursive deterministic algorithm which improves the results reported in the literature. It constructs a solution out of “perfect” or, at times, “perforated” series of integers using arbitrary rules. We call it the Deterministic Weak Schur Series (DWSS) algorithm. When analyzing our algorithm, we use the Gros sequence to bound from above the number of operations it executes. Consequently, we prove that its complexity is in O(33K). With DWSS, we discovered larger weakly sum-free partitions, breaking the literature records for K=8 to K=12.We next hybridize DWSS by stringing it together with a Monte Carlo-based method which explores the neighborhood of the solution obtained deterministically. We call it the Randomized Weak Schur Series (RWSS). As this algorithm requires more computational effort than DWSS, we are not able to compute partitions for K>10. We exceed our own DWSS results for K=8 to K=10.