A basic result in Ramsey theory states that any tournament contains a “large” transitive subgraph. Since transitive tournaments contain only transitive subgraphs, it is natural to ask which subgraphs must appear in any large tournament that is “far” from being transitive. One result of this type was obtained by Fox and Sudakov who characterized the tournaments that appear in any tournament that...

Arrow’s impossibility theorem [1] shows that the set of acyclic tournaments is not closed to non dictatorial Boolean aggregation. In this paper we extend the notion of aggregation to general tournaments and we show that for tournaments with four vertices or more any proper symmetric (closed to vertex permutations) subset cannot be closed to non dictatorial monotone aggregation and to non neutra...

We survey results on paths, trees and cycles in tournaments. The main subjects are hamiltonian paths and cycles, vertex and arc disjoint paths with prescribed endvertices, arc-pancyclicity, oriented paths, trees and cycles in tournaments. Several unsolved problems are included.

Coloring graphs is an important algorithmic problem in combinatorics with many applications in computer science. In this paper we study coloring tournaments. A chromatic number of a random tournament is of order Ω( n log(n)). The question arises whether the chromatic number can be proven to be smaller for more structured nontrivial classes of tournaments. We analyze the class of tournaments def...

We present a family of adaptive pairwise tournaments which are provably robust against large error fractions when used to determine the largest element in a set. These tournaments use nk pairwise comparisons but have only O(k + log n) depth where n is the number of players and k is a robustness parameter (for reasonable values of n and k). We show how these tournaments can be used to prove mult...

Choice functions on tournaments always select the maximal element (Condorcet winner), provided they exist, but this property does not hold in the more general case of weak tournaments. In this paper we analyze the relationship between the usual choice functions and the set of maximal elements in weak tournaments. We introduce choice functions selecting maximal elements, whenever they exist. Mor...

Department of Natural Resources (WDNR) to evaluate the mortality associated with culling in bass tournaments. The objective of this study was to quantify mortality rates of largemouth bass (Micropterus salmoides) (LMB) and smallmouth bass (M. dolomieu) (SMB) occurring as a result of tournament-induced stress, and in particular culling. Two parallel studies were conducted. The first study quanti...

L. Volkmann, Discrete Math. 245 (2002) 19-53 posed the following question. Let 4 ≤ m ≤ n. Are there strong n-partite tournaments, which are not themselves tournaments, with exactly n − m + 1 cycles of length m? We answer this question in affirmative. We raise the following problem. Given m ∈ {3, 4, . . . , n}, find a characterization of strong n-partite tournaments having exactly n −m + 1 cycle...