43 research outputs found
Positive independence densities of finite rank countable hypergraphs are achieved by finite hypergraphs
The independence density of a finite hypergraph is the probability that a
subset of vertices, chosen uniformly at random contains no hyperedges.
Independence densities can be generalized to countable hypergraphs using
limits. We show that, in fact, every positive independence density of a
countably infinite hypergraph with hyperedges of bounded size is equal to the
independence density of some finite hypergraph whose hyperedges are no larger
than those in the infinite hypergraph. This answers a question of Bonato,
Brown, Kemkes, and Pra{\l}at about independence densities of graphs.
Furthermore, we show that for any , the set of independence densities of
hypergraphs with hyperedges of size at most is closed and contains no
infinite increasing sequences.Comment: To appear in the European Journal of Combinatorics, 12 page
New lower bounds for the independence number of sparse graphs and hypergraphs
We obtain new lower bounds for the independence number of -free graphs
and linear -uniform hypergraphs in terms of the degree sequence. This
answers some old questions raised by Caro and Tuza \cite{CT91}. Our proof
technique is an extension of a method of Caro and Wei \cite{CA79, WE79}, and we
also give a new short proof of the main result of \cite{CT91} using this
approach. As byproducts, we also obtain some non-trivial identities involving
binomial coefficients