#### Title

Even 2x2 Submatrices of a Random Zero-One Matrix

#### Document Type

Article

#### Publication Date

11-1-2004

#### Description

Consider an m x zero-one matrix A. An s x t submatrix of A is said to be even if the sum of its entries is even. In this paper, we focus on the case m = n and s = t = 2. The maximum number M(n) of even 2 x 2 submatrices of A is clearly ( 2n) 2, and corresponds to the matrix A having, e.g., all ones (or zeros). A more interesting question, motivated by Turán numbers and Hadamard matrices, is that of the minimum number m(n) of such matrices. It has recently been shown that m(n) ≥ 1/2 ( 2n) 2 - Bn 3 for some constant B. In this paper we show that if the matrix A = A n is considered to be induced by an infinite zero one matrix obtained at random, then P(E n ≤1/2( 2n) 2 - Cn 2 log n infinitely often) = 0, where E n denotes the number of even 2 x 2 submatrices of A n. Results such as these provide us with specific information about the tightness of the concentration of E n around its expected value of 1/2 ( 2n) 2.

#### Citation Information

Godbole, Anant P.;
and
Johnson, Joseph A..
2004.
Even 2x2 Submatrices of a Random Zero-One Matrix.
*Graphs and Combinatorics*.
Vol.20(4).
457-466.
https://doi.org/10.1007/s00373-004-0585-9
ISSN: 0911-0119