Abstract
Tournament embedding is an order relation on the class of finite tournaments. An antichain is a set of finite tournaments that are pairwise incomparable in this ordering. We say an antichain \(\mathcal{A}\) can be extended to an antichain \(\mathcal{B}{\text{ if }}\mathcal{A} \subseteq \mathcal{B}\). Those finite antichains that can not be extended to antichains of arbitrarily large finite cardinality are exactly those that contain a member of each of four families of tournaments. We give an upper bound on the cardinality of extensions of such antichains. This bound depends on the maximum order of the tournaments in the antichain.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Cameron, P.: Orbits of permutation groups on unordered sets II, J. London Math. Soc. 23 (1981), 249–264.
Cherlin, G. L.: Combinatorial problems connected with finite homogeneity, In: Contemp. Math. 131, Amer. Math. Soc., Providence, 1992, pp. 3–30.
Cherlin, G. L. and Latka, B. J.: Minimal Antichains in well-founded quasi-orders with an application to tournaments, J. Combin. Theory B 80 (2000), 258–276.
Cherlin, G. L. and Latka, B. J.: A decision problem involving tournaments, DIMACS Technical Report 96-11, 1996.
Ding, G., Oporowski, B. and Oxley, J.: On infinite antichains of matroids, J. Combin. Theory B 63 (1995), 21–40.
Ding, G.: Subgraphs and well-quasi-ordering, J. Graph Theory 16(5) (1992), 489–502.
Ding, G.: e-mail correspondence.
Erdös, P. and Moser, L.: A problem on tournaments, Canad. Math. Bull. 7 (1964), 351–356.
Gustedt, J.: Finiteness theorems for graphs and posets obtained by compositions, Order 15 (1999), 203–220.
Henson, C. W.: Countable homogeneous relational systems and categorical theories, J. Symb. Logic 37 (1972), 494–500. MR 48 #94.
Jenkyns, T. A. and Nash-Williams, C. St. J. A.: Counterexamples in the theory of well-quasiordered sets, In: Proc. 2nd Ann Arbor Graph Theory Conf., 1968, pp. 87–91.
Klavžar, S. and Petkovšek, M.: On characterizations with forbidden subgraphs, In: Combinatorics, North-Holland, Amsterdam, 1988, pp. 331–339.
Knuth, D.: Axioms and Hulls, Lecture Notes in Comput. Sci. 606, Springer-Verlag, Berlin, 1992.
Lachlan, A. H.: Countable homogeneous tournaments, Trans. Amer. Math. Soc. 284 (1984), 431–461. MR 85i:05118.
Latka, B. J.: A classification of antichains of finite tournaments, DIMACS Technical Report 2002-24, 2002.
Latka, B. J.: Finitely constrained classes of homogeneous directed graphs, J. Symb. Logic 59 (1994), 124–139. MR 95d:03065.
Latka, B. J.: Some classes of tournaments defined by obstructions, Congressus Numerantium 104 (1994), 81–87.
Latka, B. J.: Structure theorem for tournaments omitting N 5, J. Graph Theory 42 (2003), 165–192.
Latka, B. J.: Structure theorem for tournaments omitting IC 3 (I, I,L 3 ), to appear in J. Graph Theory.
Latka, B. J.: Tournaments, embeddings, and well-quasi-orderings, In: Graph Theory, Combinatorics, and Applications, Proc. 7th Quadrennial International Conference on the Theory and Applications of Graphs, Vol. 2, 1995, pp. 685–696. MR 95d:03065.
Moon, J. W.: Tournaments whose subtournaments are irreducible or transitive, Canad. Math. Bull. 21 (1979), 75–79. MR 95d:03065.
Robertson, N. and Seymour, P. D.: Graph minors #x2013; a survey, In: Surveys in Combinatorics, 1985, London Math. Soc. Lecture Notes Ser. 103, Cambridge Univ. Press, 1985, pp. 155–171.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Latka, B.J. Antichains of Bounded Size in the Class of Tournaments. Order 20, 109–119 (2003). https://doi.org/10.1023/B:ORDE.0000009244.53720.af
Issue Date:
DOI: https://doi.org/10.1023/B:ORDE.0000009244.53720.af