Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Thursday, February 21, 2019, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Igor Balla

The Minrank of Random Graphs over Arbitrary Fields

The minrank of a graph G on the set of vertices {1, ..., n} over a field F is the minimum possible rank of an n x n matrix M over F, with nonzero diagonal entries such that M_{i,j} = 0 whenever ij is not an edge of G. We show that the minrank of the random graph G(n,p) over any field F is on the order of n log(1/p) / log(n) with high probability. For the field of real numbers, this 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 Ronyai, Babai, and Ganapathy for the number of zero-patterns of a sequence of polynomials. Joint work with Noga Alon, Lior Gishboliner, Adva Mond, and Frank Mousset.

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