Even 2x2 Submatrices of a Random Zero-One Matrix
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.
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