On four color monochromatic sets with nondecreasing diameter (Q1772414)

The author investigates a generalized Ramsey question. The function \(f(n,k)\) denotes the smallest integer \(M\) such that for every \(k-\)coloring of positive integers not exceeding \(M\), there exist two monochromatic sets \(X_1,X_2\subseteq [1,M]\), \(| X_1| =| X_2| =n,\) and the condition \(\max(X_1)<\min(X_2)\) holds. It is proved that \(f(n,4)\leq 12n-9.\) Since a classical result of Erdős, Ginzburg and Ziv implies \(f(n,4)\geq 12n-9,\) this bound is sharp. The determination of the function \(f(n,k)\) is related to a generalization of a zero-sum problem.