Independent transversals for countable set systems
نویسندگان
چکیده
منابع مشابه
A Criterion for the Existence of Transversals of Set Systems
Nash-Williams [6] formulated a condition that is necessary and sufficient for a countable family A = (Ai)i∈I of sets to have a transversal. In [7] he proved that his criterion applies also when we allow the set I to be arbitrary and require only that ⋂ i∈J Ai = ∅ for any uncountable J ⊆ I. In this paper, we formulate another criterion of a similar nature, and prove that it is equivalent to the ...
متن کاملOdd Independent Transversals are Odd
We put the final piece into a puzzle first introduced by Bollobás, Erdős and Szemerédi in 1975. For arbitrary positive integers n and r we determine the largest integer ∆ = ∆(r, n), for which any r-partite graph with partite sets of size n and of maximum degree less than ∆ has an independent transversal. This value was known for all even r. Here we determine the value for odd r and find that ∆(...
متن کاملMenger's Theorem for a Countable Source Set
Paul Erd} os has conjectured that Menger's theorem extends to innnite graphs in the following way: whenever A; B are two sets of vertices in an innnite graph, there exist a set of disjoint A{B paths and an A{B separator in this graph so that the separator consists of a choice of precisely one vertex from each of the paths. We prove this conjecture for graphs that contain a set of disjoint paths...
متن کاملIndependent transversals in r-partite graphs
Let G(r, n) denote the set of all r-partite graphs consisting of n vertices in each partite class. An independent transversal of G ∈ G(r, n) is an independent set consisting of exactly one vertex from each vertex class. Let ∆(r, n) be the maximal integer such that every G ∈ G(r, n) with maximal degree less than ∆(r, n) contains an independent transversal. Let Cr = limn→∞ ∆(r,n) n . We establish...
متن کاملOn Clique-Transversals and Clique-Independent Sets
A clique-transversal of a graph G is a subset of vertices intersecting all the cliques of G. A cliqueindependent set is a subset of pairwise disjoint cliques of G. Denote by τC(G) and αC(G) the cardinalities of the minimum clique-transversal and maximum clique-independent set of G, respectively. Say that G is clique-perfect when τC(H) = αC(H), for every induced subgraph H of G. In this paper, w...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal of Combinatorial Theory, Series A
سال: 1975
ISSN: 0097-3165
DOI: 10.1016/s0097-3165(75)80002-1