Ranks of matrices with few distinct entries

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 $L\subset \mathbb{Z}$, 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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