1. Introduction. Let a 1 2 , …, m be set of real non-negative numbers and let 1.1 P (x) = x + … (am ≠ 0). Many combinatorial problems can reduced to the study B n generated by 1.2 . Some this type were treated Touchard (7), Jacobsthal (3), Chowla, Herstein, Moore Scott (1; 2), present authors (4).

G ( n;l ) will denote a graph of n vertices and l edges. Let f 0 n, k be the smallest integer such that there is n;f (n, )) in which for every set vertex joined to each these. Thus example o = 3 since triangle pair third. It can readily checked 5 (the extremal consists complete 4-gon with one edge removed). In general we prove: > , then f(n, k) .

(1966). The Theory of Round Robin Tournaments. American Mathematical Monthly: Vol. 73, No. 3, pp. 231-246.

1. Introduction. Several recent papers have dealt with the number of solutions x d = 1 in S n , symmetric group degree . Let us denote this by A n,d and let n,2 T

(1952). On The Different Distances Determined By n Points. American Mathematical Monthly: Vol. 59, No. 2, pp. 85-91.

Let 5 be a set of positive integers no three which are in arithmetical progression, i.e., if A, B, C distinct elements S, A + B ≠ 2C. We call such non-averaging set. v(n) denote the maximum number not exceeding n any The problem finding bounds for has been treated by several authors [1, 3, 5, 6, 7]. question first arose connection with theorem van der Waerden [8].

In a previous paper (1) the authors considered problem of finding an asymptotic formula for numbers or functions B n,m whose generating function is form (1.1) , where P m ( x ) polynomial degree in given by (1.2) ≠0. The above-mentioned contained restriction that k ≥ 0.

Two teams A and B can be compared by matching each player in against B. We say that > if only the players of collectively win more games than they lose.If there are n T lf 2 , then outcomes matches between various may represented an oriented graph H on nodes which arc goes from i-th node to j-th T; Tj.It is shown any generated this way, minimum number necessary generate order z llog n.

Previous article Next On the Correlation Function of Random Binary SequencesJ. W. Moon and L. MoserJ. Moserhttps://doi.org/10.1137/0116028PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] R. H. Barker, Group synchronizing binary digital systems, Communication Theory, London, 1953, 273–287 Google Scholar[2] P. Dilworth, Note on strong law large numbers, Amer. Math. Monthly, 56 (1949), 249–250 MR0030146 (10,720d) 0032.41604 CrossrefGoogle Scholar[3] S....

Given a set of n points, with each pair distinct points joined by line that is oriented towards exactly one the then resulting configuration called (roun-drobin) tournament. A tournament reducible if can be separated into two non-empty subsets, and B, such every joins point in to B B. If not it irreducible. The object this note derive an approximation for P(n), probability on point, chosen at random from possible ones, will p(1)=1, definition.

A well known theorem of van der Waerden [1] states that given any two positive integers k and t, there exists a integer m such in every distribution the numbers 1,2, …, into classes, at least one class contains an arithmetic progression t + 1 terms. Other proofs generalizations this have been by Griinwald [2], Witt [3] Lukomskaya [4]. The last mentioned proof appears booklet Khinchin “Three pearls number theory” which Waerden's plays role first pearl.

By a tournament we mean the outcome of round-robin in which there are no draws. Such may be represented by graph n players vertices labelled 1, 2, …, n, and outcomes games directed edges so that every pair is joined one edge. We call such complete graph. One can also represent an nXn matrix T=(t ij ) t 1 if i beats j, 0 otherwise, T (0, 1) with + ji = for i≠j (by definition) ii 0.