On Minrank and Forbidden Subgraphs

The minrank over a field F of a graph G on the vertex set { 1,2,… ,n} is the minimum possible rank of a matrix M ∈ Fn × n such that Mi, i ≠ 0 for every i, and Mi, j =0 for every distinct non-adjacent vertices i and j in G. For an integer n, a graph H, and a field F, let g(n,H, F) denote the maximum possible minrank over F of an n-vertex graph whose complement contains no copy of H. In this article, we study this quantity for various graphs H and fields F. For finite fields, we prove by a probabilistic argument a general lower bound on g(n,H,F), which yields a nearly tight bound of Ω (√ n/ log n) for the triangle H=K3. For the real field, we prove by an explicit construction that for every non-bipartite graph H, g(n,H, R) ≥ nδ for some δ = δ (H)> 0. As a by-product of this construction, we disprove a conjecture of Codenotti et al. [11]. The results are motivated by questions in information theory, circuit complexity, and geometry.