Some intersection theorems

Abstract

LetL(A) be the set of submatrices of anm×n matrixA. ThenL(A) is a ranked poset with respect to the inclusion, and the poset rank of a submatrix is the sum of the number of rows and columns minus 1, the rank of the empty matrix is zero. We attack the question: What is the maximum number of submatrices such that any two of them have intersection of rank at leastt? We have a solution fort=1,2 using the followoing theorem of independent interest. Letm(n,i,j,k) = max(|F|;|G|), where maximum is taken for all possible pairs of families of subsets of ann-element set such thatF isi-intersecting,G isj-intersecting andF ansd,G are cross-k-intersecting. Then fori≤j≤k, m(n,i,j,k) is attained ifF is a maximali-intersecting family containing subsets of size at leastn/2, andG is a maximal2k−i-intersecting family.

Furthermore, we discuss and Erdős-Ko-Rado-type question forL(A), as well.

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Research was (partially) supported by Hungarian National Foundation for Scientific Research grant no. 1812

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Sali, A. Some intersection theorems. Combinatorica 12, 351–361 (1992). https://doi.org/10.1007/BF01285823

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AMS subject classification code (1991)

  • 05 A 05
  • 06 A 10