Palindromic optimal colorings for Schur numbers

Determine whether, for every integer k ≥ 1, there exists an optimal k‑coloring achieving the Schur number S(k) that is palindromic under index reversal i ↦ n−i+1; that is, an optimal coloring of {1,…,S(k)} with no monochromatic solution to x+y=z that is symmetric with respect to reversing the index order.

Background

The Schur number S(k) is the largest n such that there exists a k‑coloring of {1,…,n} without a monochromatic solution to x+y=z. Known optimal solutions for small k exhibit palindromic symmetry (colorings invariant under reversal i ↦ n−i+1). While symmetry often aids computational search and human understanding, it is unknown whether every optimal Schur coloring can be chosen palindromic for all k. The paper cites this as a motivating example of symmetry in combinatorial constructions.