# 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.

This is a preview of subscription content, access via your institution.

## References

1. [1]

P. Erdős, Chao Ko andR. Rado: Intersection theorems for systems of finite sets,Quart. J. Math. Oxford (2)12 (1961), 313–320.

2. [2]

P. Frankl: On intersecting families of finite sets,J. of Combin. Theory (A) 24 (1978), 146–161.

3. [3]

P. Frankl: The shifting technique in extremal set theory,Combinatorial Surveys, 1987 (C. Whitehead ed.) Cambridge Univ. Press, 1987, 81–110.

4. [4]

G.O.H. Katona: Intersection theorems for systems of finite sets,Acta Math. Acad. Sci. Hung. 15 (1964), 329–337.

5. [5]

L. Lovász:Combinatorial Problems and Exercises, Akadémiai Kiadó, Budapest North Holland Publ., Amsterdam 1979.

6. [6]

L. Pyber: A new generalization of the Erdős-Ko-Rado Theorem,J. of Combin. Theory (A) 43, 85–90.

7. [7]

A. Sali: Extremal Theorems for Submatrices of a Matrix,Colloq. Math. Soc. János Bolyai 52, Combinatorics, Eger (Hungary), 1987, 439–446.

## Author information

### Affiliations

Authors

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

## Rights and permissions

Reprints and Permissions

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