Abstract
An L-matrix is a matrix whose off-diagonal entries belong to a set L, and whose diagonal is zero. Let N(r, L) be the maximum size of a square L-matrix of rank at most r. Many applications of linear algebra in extremal combinatorics involve a bound on N(r, L). We review some of these applications, and prove several new results on N(r, L). In particular, we classify the sets L for which N(r, L) is linear, and show that if N(r, L) is superlinear and L ⊂ Z, then N(r, L) is at least quadratic.
As a by-product of the work, we asymptotically determine the maximum multiplicity of an eigenvalue λ in an adjacency matrix of a digraph of a given size.
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In memory of a great teacher, Jirka Matoušek
Supported in part by U.S. taxpayers via NSF grant DMS-1301548.
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Bukh, B. Ranks of matrices with few distinct entries. Isr. J. Math. 222, 165–200 (2017). https://doi.org/10.1007/s11856-017-1586-8
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DOI: https://doi.org/10.1007/s11856-017-1586-8