An Erdös-Ko-Rado theorem for the subcubes of a cube

Abstract

LetP be that partially ordered set whose elements are vectors x=(x 1, ...,x n ) withx i ε {0, ...,k} (i=1, ...,n) and in which the order is given byxy iffx i =y i orx i =0 for alli. LetN i (P)={x εP : |{j:x j ≠ 0}|=i}. A subsetF ofP is called an Erdös-Ko-Rado family, if for allx, y εF it holdsxy, x ≯ y, and there exists az εN 1(P) such thatzx andzy. Let ℓ be the set of all vectorsf=(f 0, ...,f n ) for which there is an Erdös-Ko-Rado familyF inP such that |N i (P) ∩F|=f i (i=0, ...,n) and let 〈ℓ〉 be its convex closure in the (n+1)-dimensional Euclidean space. It is proved that fork≧2 (0, ..., 0) and\(\left( {0,...,0,\overbrace {i - component}^{\left( {\begin{array}{*{20}c} {n - 1} \\ {i - 1} \\ \end{array} } \right)}k^{i - 1} ,0,...,0} \right)\) (i=1, ...,n) are the vertices of 〈ℓ〉.

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Engel, K. An Erdös-Ko-Rado theorem for the subcubes of a cube. Combinatorica 4, 133–140 (1984). https://doi.org/10.1007/BF02579213

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AMS subject classification (1980)

  • 05 A 99
  • 06 A 10