Abstract
For a hypergraphH, we denote by
-
(i)
τ(H) the minimumk such that some set ofk vertices meets all the edges,
-
(ii)
ν(H) the maximumk such that somek edges are pairwise disjoint, and
-
(iii)
λ(H) the maximumk≥2 such that the incidence matrix ofH has as a submatrix the transpose of the incidence matrix of the complete graphK k .
We show that τ(H) is bounded above by a function of ν(H) and λ(H), and indeed that if λ(H) is bounded by a constant then τ(H) is at most a polynomial function of ν(H).
Similar content being viewed by others
References
D. Bienstock, andN. Dean: Some obstructions to small face covers in planar graphs,J. Combinatorial Theory, (B), to appear.
A. Blumer, A. Ehrenfeucht, D. Haussler, andM. K. Warmuth: Learnability and the Vapnik-Čhervonenkis dimension,J. Assoc. Comp. Mach. 36 (1989), 929–965.
J. Folkman: Notes on the Ramsey numberN (3,3,3,3),J. Combinatorial Theory, (A),16 (1974), 371–379.
D. Haussler, andE. Welzl: ε-nets and simplex range queries,Discrete Comput. Geom. 2 (1987), 127–151.
J. Komlós, J. Pach, andG. Woeginger: Almost tight bounds for epsilon-nets,Disc. and Comput. Geom.,7 (1992), 163–173.
J. Pach, andJ. Törőcsik: Some geometric applications of Dilworth's Theorem,Proc. 9th ACM Symposium on Comput Geom. 1993.
E. Slud: Distribution inequalities for the binomial law,Annals of Probability 5 (1977), 404–412.
V. N. Vapnik, andA. Ya. Chervonenkis: On the uniform convergence of relative frequencies of events to their probabilities,Theory Probab. Appl. 16 (1971), 264–280.