Number of Cliques in Graphs with a Forbidden Subdivision

Abstract

We prove that for all positive integers t, every n-vertex graph with no K[subscript t]-subdivision has at most 2[superscript 50t]n cliques. We also prove that asymptotically, such graphs contain at most 2[superscript (5+o(1))t]n cliques, where o(1) tends to zero as t tends to infinity. This strongly answers a question of Wood that asks whether the number of cliques in n-vertex graphs with no K[subscript t]-minor is at most 2[superscript ct]n for some constant c.