By B. Bollobás (Eds.)
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However, let us 40 B. Bollobas, P. Erdos, M. Simonovits, E. Szemeridi apply the Erdos theorem again, now to the hypergraph whose vertices are in X , U X,U. . UX, and the hyperedges of which are some cycles of G" of form ( x , ~ ,. . , x,,),x, E X,, where we choose only one permutation i,, . . If. , . , x,)defines a cycle in G" and I Y,I = t. Thus we obtained a C 1 ( tc) G". g. V,, . . , V, define a shortest odd cycle in Rk,by the assumption that the edges are regularly distributed in G", there must be a V,, q > j, which is joined to at least 2c'j of the classes V,, .
UX, and the hyperedges of which are some cycles of G" of form ( x , ~ ,. . , x,,),x, E X,, where we choose only one permutation i,, . . If. , . , x,)defines a cycle in G" and I Y,I = t. Thus we obtained a C 1 ( tc) G". g. V,, . . , V, define a shortest odd cycle in Rk,by the assumption that the edges are regularly distributed in G", there must be a V,, q > j, which is joined to at least 2c'j of the classes V,, . . , V,. If V, is joined to a V, and V,. for some i t farther from z than 2, then the arc V,V,V,.
By the well known Bernstein inequality for binomial distributions (see p. 387 in ) the probability that U is joined to more than 22'- ' c ' n + n2'3 or to less than 22rp'crn- n2'3 vertices x E B completely is 0 (exp ( - cln"')). Hence with probability tending to 1 on each U there is a K2(r,t ) for t = 2 2 r p ' c r n n213 but on no U for t = 2 2 r - 1 ~ + r nn2/', since A similar application of Bernstein's inequality yields that l e ( G " ) - cn'l s n5/' with probability tending to 1. Exactly the same argument gives that if G" is a random subgraph of K, of size [en]' (or is obtained from K,, by choosing each edge with probability 2 c ) then G" will contain a K2(r, t ) for t = 2'c'n - nZi3with probability tending to one, but for t = 2'c'n + n213only with probability tending to 0.