# Download Advances in Graph Theory by B. Bollobás (Eds.) PDF

By B. Bollobás (Eds.)

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Simonovits, E. Szemeridi (iv) Let Bibe the set of vertices in Ai joined to at least cn vertices of the same class Ai. Then lBil< M. Proof. Let M,, = R and choose natural numbers M , < M2<. Put M = Md. Pick q such that 0 < q < (tc)". By Lemma 5 (ii) we can choose E , 0 < E < c, and n, such that if N = [ q n ] ,n 2 n , and in H = Gd(N,N , . . , N ) at most &n2 edges are missing between any two classes then H contains a Kd(R,R, . . , R). 2 in [I]) implies that there exist noZ n , and 6 > 0 with the following properties.

The first assertion is an immediate consequence of Lemma 5. Instead of the second we prove the following stronger assertion. ) with probability 4c. Then, with probability tending to 1, G" has cn2+o(n2)edges and if t = t ( G " )is the maximal number for which G" contains a K2(r, t ) then, again with probability tending to 1, we have c'n + o ( n ) . In order to prove this assertion, we denote by A and B the two classes of K&n, \$n). Bollobus, P. Erdos, M. Simonovits, E. Szemeridi 36 and x is joined to every vertex in U.

Simonouits, E. Szemerkdi where a = /1 = rIJs s + , V , j = ( s + l ) m . To obtain an upper bound of d in terms of a, we apply Lemma 5 to the bipartite graph determined by the classes Uzs,ss+lV , (=first class) and V, (=second class). We find that G" I K z ( r , t ) with t = ( 1-o(l))drnra-(r-l1. (7) By the assumption G " 3Kz(r, 22'C'c;n) and by (7) (8) drnrC1a-(rCIJ< ( 1+0(1))2~'-'c;. Let us assume that d > 2c, (this will be shown later). From (8) and c;