New lower bounds for weak Schur partitions

TL;DR

This paper establishes new lower bounds for the partitioning of integer intervals into weakly sum-free subsets, demonstrating that their growth rate exceeds that of strongly sum-free partitions, with specific bounds provided for small numbers of subsets.

Contribution

It introduces a novel method related to Schur's original approach to derive new lower bounds for weak Schur partitions, extending the understanding of their asymptotic growth.

Findings

Lower bounds for weak Schur partitions are significantly improved.

The growth rate of weak Schur partitions exceeds 3.27 asymptotically.

Explicit bounds are provided for partitions into 6 to 10 subsets.

Abstract

This paper records some apparently new results for the partition of integer intervals [1, n] into weakly sum-free subsets. These were produced using a method closely related to that used by Schur in 1917. New lower bounds can be produced in this way for partitions of unlimited size. The asymptotic growth rate of the lower bounds, as the number of subsets increases, cannot be less than the same growth rate for strongly sum-free partitions, and therefore exceeds 3.27. Specific results for partitions into a 'small' number of subsets include $WS(6) \ge 642$, $WS(7) \ge 2146$, $WS(8) \ge 6976$, $WS(9) \ge 21848$, and $WS(10) \ge 70778$.