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 ∈ F n × n such that M i, i ≠ 0 for every i , and M i, 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 = K 3 . For the real field, we prove by an explicit construction that for every non-bipartite graph H , g ( n , H , R) ≥ n δ 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.