On the maximum number of cliques in a graph embedded in a surface

This paper studies the following question: given a surface Σ and an integer n , what is the maximum number of cliques in an n -vertex graph embeddable in Σ ? We characterise the extremal graphs for this question, and prove that the answer is between 8 ( n − ω ) + 2 ω and 8 n + 5 2 2 ω + o ( 2 ω ) , where ω is the maximum integer such that the complete graph K ω embeds in Σ . For the surfaces S 0 , S 1 , S 2 , N 1 , N 2 , N 3 and N 4 we establish an exact answer.