Hook-free colorings and a problem of Hanson

Abstract

Hanson posed the following problem: What is the minimum numberχ(n) of colors needed to color all subsets of ann-set such that there is no monochromatic tripleA, B, C withA ∪B=C? It is known thatχ(n)≦[(n+1)/2], while Erdős and Shelah provedχ(n)≧[(n+1)/4]. Their proof suggests the following notion: LetC be any finite plane point-configuration. The hook-free coloring numberχ(C) is the smallest number of colors needed forC such that no monochromatic hooks arise, i.e. if (c _x ,c _y ) are the coordinates of pointc∈C, then there are no 3 distinct pointsa, b, c∈C witha _x =b _x <c _x ,b _y =c _y <a _y . In this paperχ(R _m,n ) is determined exactly for anm×n-rectangle, and asymptotically for the triangular staircase. As a corollary one obtainsχ(n)≧0.293n.