The Minrank of Random Graphs over Arbitrary Fields
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The minrank of a graph G on the set of vertices [n] over a field F is the minimum possible rank of a matrix M∈F^n× n with nonzero diagonal entries such that M_i,j=0 whenever i and j are distinct nonadjacent vertices of G. This notion, over the real field, arises in the study of the Lovász theta function of a graph. We obtain tight bounds for the typical minrank of the binomial random graph G(n,p) over any finite or infinite field, showing that for every field F= F(n) and every p=p(n) satisfying n^-1≤ p ≤ 1-n^-0.99, the minrank of G=G(n,p) over F is Θ(n (1/p)/ n) with high probability. The result for the real field settles a problem raised by Knuth in 1994. The proof combines a recent argument of Golovnev, Regev, and Weinstein, who proved the above result for finite fields of size at most n^O(1), with tools from linear algebra, including an estimate of Rónyai, Babai, and Ganapathy for the number of zero-patterns of a sequence of polynomials.
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