A problem of Erdős on the minimum number of k-cliques

Fifty years ago Erdős asked to determine the minimum number of k-cliques in a graph on n vertices with independence number less than l. He conjectured that this minimum is achieved by the disjoint union of l− 1 complete graphs of size n l−1 . This conjecture was disproved by Nikiforov who showed that the balanced blow-up of a 5-cycle has fewer 4-cliques than the union of 2 complete graphs of size n2 . In this paper we solve Erdős’ problem for (k, l) = (3, 4) and (k, l) = (4, 3). Using stability arguments we also characterize the precise structure of extremal examples, confirming Erdős’ conjecture for (k, l) = (3, 4) and showing that a blow-up of a 5-cycle gives the minimum for (k, l) = (4, 3).