On Extremal Problems of Graphs and Generalized Graphs

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 ; Gt'1(n ;(n)), the complete r-graph of n vertices, r will be denoted by K(')(n), i .e ., Kt' 1(n) contains all the r-tuples formed from n elements . Kt' 1(n 1 , . . ., n,) will denote the r-graph of E = 1 nivertices andfj= 1 ni r-tuples defined as follows : The vertices are x;'1, :5 r, 1 ' . . . x~' 1 ), 1 < i,< nj , 1 < j < r. Thus K(2)(2,2) is simply a rectangle. Denote byf(')(n) the smallest integer so that every G(')(n ;f(')) contains a complete r-graph of l vertices . As is well known Turán [5] determined f'(2)(n) for every l and n and also proved that there is a unique G (2) (n ;ff 2)(n) 1) which contains no complete 2-graph of l vertices (ordinary graphs have to be denoted as 2-graphs here). In particular f32 (n) = [n'/4] + 1 . For r > 2 the determination of f( "'( n) seems to be a very difficult question which is unsolved for all r > 2, 1 > r . (This question was also posed by Turán . Turán in particular conjectured that (1) Received August 18, 1964 . f 5 3)(n) = n 2(n 1) .