Ramseyan variations on symmetric subsequences

Abstract

A theorem of Dekking in the combinatorics of words implies that there exists an injective order-preserving transformation \(f : {\{0,1,\ldots,n\}}\rightarrow  {\{0,1,\ldots,2n\}}\) with the restriction \(f(i+1)\le f(i) + 2\) such that for every 5-term arithmetic progression \(P\) its image \(f(P)\) is not an arithmetic progression. In this paper we consider symmetric sets in place of arithmetic progressions and prove lower and upper bounds for the maximum \(M=M(n)\) such that every \(f\) as above preserves the symmetry of at least one symmetric set \(S\subseteq\{0,1,\ldots,n\}\) with \(|S|\ge M\).