On the number of cliques in graphs with a forbidden minor

Reed and Wood and independently Norine, Seymour, Thomas, and Wollan proved that for each positive integer $t$ there is a constant $c(t)$ such that every graph on $n$ vertices with no $K_t$-minor has at most $c(t)n$ cliques. Wood asked in 2007 if we can take $c(t) = c^t$ for some absolute constant $c$. This question was recently answered affirmatively by Lee and Oum. In this paper, we determine the exponential constant. We prove that every graph on $n$ vertices with no $K_t$-minor has at most $3^{2t/3+o(t)}n$ cliques. This bound is tight for $n \geq 4t/3$. More generally, let $H$ be a connected graph on $t$ vertices, and $x$ denote the size (i.e., the number edges) of the largest matching in the complement of $H$. We prove that every graph on $n$ vertices with no $H$-minor has at most $\max(3^{2t/3-x/3+o(t)}n,2^{t+o(t)}n)$ cliques, and this bound is tight for $n \geq \max (4t/3-2x/3,t)$ by a simple construction. Even more generally, we determine explicitly the exponential constant for the maximum number of cliques an $n$-vertex graph can have in a minor-closed family of graphs which is closed under disjoint union.

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