On a problem of Duke-Erdös-Rödl on cycle-connected subgraphs

A note on Erdös’s conjecture on multiplicities of complete subgraphs

, and let ct = limn→∞ ct(n). An old conjecture of Erdös, related to Ramsey’s theorem, states that ct = 2 1−(t 2 ). It was shown false by Thomason for all t ≥ 4 ([3],[4]). Franek and Rödl ([1]) presented a simpler counterexample to the conjecture for t = 4 derived from a simple Cayley graph of order 2 obtained by a computer search giving essentially the same upper bound for c4 as Thomason’s. In ...

An algorithmic hypergraph regularity lemma

Szemerédi’s Regularity Lemma [22, 23] is a powerful tool in graph theory. It asserts that all large graphs G admit a bounded partition of E(G), most classes of which are bipartite subgraphs with uniformly distributed edges. The original proof of this result was non-constructive. A constructive proof was given by Alon, Duke, Lefmann, Rödl and Yuster [1], which allows one to efficiently construct...

Ramsey problem on multiplicities of complete subgraphs in nearly quasirandom graphs

Let kt(G) be the number of cliques of order t in the graph G. For a graph G with n vertices let ct(G) = kt(G)+kt(Ḡ) (n t ) . Let ct(n) = Min{ct(G) : |G| = n} and let ct = limn→∞ ct(n). An old conjecture of Erdös [2] related to Ramsey’s theorem states that ct = 2 1−( 2). Recently it was shown to be false by A. Thomason [12]. It is known that ct(G) ∼ 21−( t 2) whenever G is a pseudorandom graph. ...

On K4-Free Subgraphs of Random Graphs

More Results on Subgraphs with Many Short Cycles

1 . Introduction In a previous paper [2] the senior authors proved the following theorem: Let G, = GI (n ;cn 2 ) be a graph of n vertices and cn 2 edges, c a positive constant . Then for n sufficiently large there is always a subgraph G 2 = G2 (m ;f(c)n 2 ) of G, every two edges of which lie together on a cycle of length at most. 6 in G 2 , and if two edges of G 2 have a common vertex they are ...