Georgia Institute of Technology College of Sciences

Fifty years ago Erdos asked to determine the minimum number of $k$-cliques in a graph on $n$ vertices with independence number less than $l$ (we will refer this as $(k,l)$-problem). He conjectured that this minimum is achieved by the disjoint union of $l-1$ complete graphs of size $\frac{n}{l-1}$. This conjecture was disproved by Nikiforov who showed that Erdos' conjecture can be true only for finite many pairs of $(k,l)$. For $(4,3)$-problem, Nikiforov further conjectured that the balanced blow-up of a $5$-cycle achieves the minimum number of $4$-cliques. We first sharpen Nikiforov's result and show that Erdos' conjecture is false whenever $k\ge 4$ or $k=3, l\ge 2074$. After introducing tools (including Flag Algebra) used in our proofs, we state our main theorems, which characterize the precise structure of extremal examples for $(3,4)$-problem and $(4,3)$-problem, confirming Erdos' conjecture for $(k,l)=(3,4)$ and Nikiforov's conjecture for $(k,l)=(4,3)$. We then focus on $(4,3)$-problem and sketch the proof how we use stability arguments to get the extremal graphs, the balanced blow-ups of $5$-cycle. Joint work with Shagnik Das, Hao Huang, Humberto Naves and Benny Sudakov.