Given a family I of independent sets in graph, rainbow set is an such that there injection ?:I?I where for each v?I, v contained ?(v). Aharoni et al. (2019) determined various graph classes C whether satisfies property every n, exists N=N(C,n) N size n contains n. In this paper, we add two dense satisfying property, namely, the class graphs bounded neighborhood diversity and r-powers expansion ...

A set Y ⊆ R that intersects every convex set of volume 1 is called a Danzer set. It is not known whether there are Danzer sets in R with growth rate O(T ). We prove that natural candidates, such as discrete sets that arise from substitutions and from cut-and-project constructions, are not Danzer sets. For cut and project sets our proof relies on the dynamics of homogeneous flows. We consider a ...

Some modifications of the definition of density of subsets in ordered (= partially ordered) sets are given and the corresponding concepts are compared.

The correspondence between perfect difference sets and transitive projective planes is well-known. We observe that all known dense (i.e., close to square-root size) Sidon subsets of abelian groups come from through a similar construction. classify the arising in this manner desarguesian find essentially no new examples, but there are many further examples nondesarguesian planes. conjecture aris...

We study the probability theory of countable dense random subsets of (uncountably infinite) Polish spaces. It is shown that if such a set is stationary with respect to a transitive (locally compact) group of symmetries then any event which concerns the random set itself (rather than accidental details of its construction) must have probability zero or one. Indeed the result requires only quasi-...