In [KW] it was prooved that for every boolean function f there exist a communication complexity game R f such that the minimal circuit-depth of f exactly equals to the communication complexity of R f . If f is monotone then there also exists a game R m f with communication complexity exactly equals to the monotone depth of f . It was also prooved in [KW] that the communication complexity of R m st\Gammaconnectivity is\Omega\Gamma/34 2 n), or eqivalently that the monotone depth of the s-t connectivity function is\Omega\Gamma/38 2 n). In this paper we consider the games R f and R m f in a probabilistic model of communication complexity, and prove that the communication complexity of R m st\Gammaconnectivity is \Omega\Gamma/20 2 n) even in the probabilistic case. We also prove that in every NC1 circuit for s-t connectivity at least a constant fraction of all input variables must be negated. 1 Introduction In standard communication complexity, two players are trying to compute...

In 1946, Erdos and Stone [3] proved that every graph with n vertices and at least edges contains a large Kd+l(t), a complete (d + l)-partite graph with t vertices in each part. More recently, Bollobas, Erdos and Simonovits [2] proved that log n t> a d log (1/c) for some positive constant a and conjectured that this bound can be improved into log n f ^ log (1/c) for some positive constant b. The purpose of our paper is to prove that log n 1 ^ 500 log (1/c) for all n large enough with respect to c and d. In a sense, this result is best possible: as shown by Bollobas and Erdos [1], our constant 1/500 cannot be increased beyond 5. Our argument hinges on a theorem asserting that every sufficiently large graph can be partitioned into a small number of classes in such a way that the partition exhibits strong regularity properties. To state the theorem in more precise terms, we need a few definitions. When A and B are nonempty disjoint sets of vertices in some graph then we denote by d{A, B) the density of edges between A and B: this is the number of edges with one endpoint in A and the other endpoint in B divided by \A\-\B\. The pair (A,B) is called E-regular if X £ A, \X \ ^ E\A\, Y < = B and |V | ^ E\B \ imply that \d(X, Y) — d(A,B) \ < E; otherwise the pair is E-irregular. A partition of a set V into classes CO,CX,...,Ck is called E-regular if |C0 | ^ F\V\, |C, | = \Cj \ whenever 1 ^ / < j ^ k and if at most F.k 2 of the pairs (ChCj) with 1 ^ / < j ^ k are <;-irregular. The Regular Partition Theorem, proved in [5], asserts that for every positive E and for every positive integer m there are positive integers M and N with the following property: if the set V of vertices of a graph has size at least N then there is an ^--regular partition of V into k+ 1 classes such that m ^ k ^ M. LEMMA 1. For every choice of positive numbers c, d and E such that d is an integer and E < c/10 there are positive numbers N and d with the following property. Ifn ^ N