Club does not imply the existence of a Suslin tree

We prove that club does not imply the existence of a Suslin tree, so answering a question of I. Juhasz

Hypernatural Numbers as Ultrafilters

In this paper we present a use of nonstandard methods in the theory of ultrafilters and in related applications to combinatorics of numbers

A coloring result for the plane

Abstract. It is possible to color the plane with countably many colors such that if H is the rational points of a line (i.e., H = ϕ[Q] for some rigid motion) then H gets every color exactly once. There are several results which show the paradoxical behavior of the Axiom of Choice by exhibiting some strange decompositions of Euclidean spaces. One such theorem is due to Erdős, who in [1] deduced from a graph theory result of his and Hajnal's [2] that the plane can be colored with countably many colors in such a way that points in rational distance get different colors. One common feature of those results (and proofs) is that what is required of the color classes is being small in some sense, so for example, we can refine color classes, if needed. In this paper we formulate and prove a result (generalizing Erdős') which states the existence of a certain coloring of the plane with countably many colors and the color classes must be small and large in the same time. For this we consider the hypergraph H in which the hyperedges are the rational points of lines. We show that there is a coloring of the plane with countably many colors such that every hyperedge of H meets every color class in exactly one point. 1991 Mathematics Subject Classification. 03E05, 04A20

On a result of Thomassen

We give a new proof of Thomassen’s theorem stating that if the chromatic (coloring) number of a graph X is > κ, then X contains a κ-edgeconnected subgraph with similar properties. © 2015 American Mathematical Society

Universal graphs without large cliques

492 revision:1993-08-22 modified:1993-08-2