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 withAB=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.

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References

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    P. Erdős, C. Ko andR. Radó, Intersection theorems for finite sets,Quart J. Math. (Oxford) (2)12 (1961), 313–318.

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Aigner, M., Grieser, D. Hook-free colorings and a problem of Hanson. Combinatorica 8, 143–148 (1988). https://doi.org/10.1007/BF02122795

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AMS subject classification (1980)

  • 05 A 05
  • 05 A 99