By B. Bollobás (Eds.)

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32 B. Bollobas, P. Erdos M . 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.

Xn-19 ~ i - 1 xn)(xn, , Y:, where i = 1 , 2 , . . , i ( n - 2 ) ; y j f y‘; for i f k and where the set { y i : i = 1 , 2 , . . ,b(n - 2) is determined as follows. Consider the $(n- 2) elements representative of the triples (x,, x,+,, y ) containing the pair {x,, x,,,} and where neither x, nor x , , ~ has been chosen. Then split these elements into two sets of x,+,) and Y’{x,, x,+,l. Then the set {y; : i = 1 , 2 , . . ,% ( n -2)} cardinality i(n - 2) : YIx,, is either the set YiX, x,+,l or Y’{x,, x,+,l according as the arc (x,, x,,,) or the arc (x,,,, x l ) appears in the directed hamiltonian cycle.

Heydemann and D. Sotteau, Hypergraphes hamiltoniens, in: Coll. Int. , Problbmes Combinatoires et Th6orie des Graphes, Orsay, 1976. [8] J. D. Keedwell, Latin squares and their applications, (Akademiai Kiado, Budapest and English University Press, London, 1974). [9] C. Huang and A. Rosa, On sets of orthogonal hamiltonian circuits, Proc. 2nd Manitoba Conference on Numerical Mathematics, Utilitas Math. Congressus Numerantium VII (1972), 327-332. D. Keedwell, Some problems concerning complete latin squares, Combinatorin, Proc.