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An r-graph is a graph whose basic elements are its vertices and r-tuples . It is proved that to every 1 and r there is an e(l, r) so that for n > no every r-graph of n vertices and n '-E(i, r) r-tuples contains r . /vertices P) ,I5 j < r, l < i < l, so that all the r-tuples (x i ,( 1 ), xi2 (2 ) ' . . . , x1 (')) occur in the r-graph.By an r-graph G (')(r >_ 2) we shall mean a graph whose basic elements are its vertices and r-tuples ; for r = 2 we obtain the ordinary graphs . These generalised graphs have not yet been investigated very much . G ( ') (n ; m) will denote an r-graph of n vertices and m r-tuples

Recently Littlewood and Offord 1 proved the following lemma: Let be complex numbers with \xi\ ^1. Consider the sums ]C*=i € &#fc> where the €& are ±1. Then the number of the sums ]Cft=i € & x * which fall into a circle of radius r is not greater than cr2 n (log n)nr 112. In the present paper we are going to improve this to cr2 n n~1 12. The case Xi = 1 shows that the result is best possible as far as the order is concerned. First we prove the following theorem. CONJECTURE. Let xi, # 2 , • • • , x n be n vectors in Hilbert space, ||#;|| èl. Then the number of sums 22=i € fc#& which f all in the interior of an arbitrary sphere of radius 1 does not exceed C n , m .

Let 0 < p < 1 be fixed and denote by G a random graph with point set , the set of natural numbers, such that each edge occurs with probability p, independently of all other edges. In other words the random variables eij, 1 ≤ i < j, defined byare independent r.v.'s with P(eij = 1) = p, P(eij = 0) = 1 − p. Denote by Gn the subgraph of G spanned by the points 1, 2, …, n. These random graphs G, Gn will be investigated throughout the note. As in (1), denote by Kr a complete graph with r points and denote by kr(H) the number of Kr's in a graph H. A maximal complete subgraph is called a clique. In (1) one of us estimated the minimum of kr(H) provided H has n points and m edges. In this note we shall look at the random variablesthe number of Kr's in Gn, andthe maximal size of a clique in Gn.

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