Abstract.
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 
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 
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 
where E
n
denotes the number of even 2×2 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 
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Acknowledgments. The research of both authors was supported by NSF Grant DMS-0139291 and conducted at East Tennessee State University during the Summer of 2002, when Johnson was a student in Godbole’s Research Experiences for Undergraduates Program. The valuable suggestions of two anonymous referees are gratefully acknowledged.
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Godbole, A., Johnson, J. Even 2 × 2 Submatrices of a Random Zero-One Matrix. Graphs and Combinatorics 20, 457–466 (2004). https://doi.org/10.1007/s00373-004-0585-9
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DOI: https://doi.org/10.1007/s00373-004-0585-9