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

This paper studies the following question: given a surface @S and an integer n, what is the maximum number of cliques in an n-vertex graph embeddable in @S? We characterise the extremal graphs for this question, and prove that the answer is between 8(n-@w)+2^@w and 8n+522^@w+o(2^@w), where @w is the maximum integer such that the complete graph K"@w embeds in @S. For the surfaces S"0, S"1, S"2, N"1, N"2, N"3 and N"4 we establish an exact answer.