Even 2 × 2 Submatrices of a Random Zero-One Matrix

Consider an m×n zero-one matrix A. An s×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 2×2 submatrices of A is clearly [inline-graphic not available: see fulltext] 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 [inline-graphic not available: see fulltext] for some constant B. In this paper we show that if the matrix A=An is considered to be induced by an infinite zero one matrix obtained at random, then [inline-graphic not available: see fulltext] where En denotes the number of even 2×2 submatrices of An. Results such as these provide us with specific information about the tightness of the concentration of En around its expected value of [inline-graphic not available: see fulltext]