Very Large Cliques are Easy to Detect * (Preliminary Version)

A Andreev, S Jukna
2006 unpublished
It is known that, for every constant k ≥ 3, the presence of a k-clique (a complete subgraph on k vertices) in an n-vertex graph cannot be detected by a monotone boolean circuit using fewer than Ω((n/ log n) k) gates. We show that, for every constant k, the presence of an (n − k)-clique in an n-vertex graph can be detected by a monotone circuit using only O(n 2 log n) gates. Moreover, if we allow unbounded fanin, then O(log n) gates are enough.