Assaf Rinot

Chromatic numbers of graphs -- large gaps

We say that a graph G is (ℵ0,κ)-chromatic if Chr(G) = κ, while Chr(G′) ≤ ℵ0 for any subgraph G′ of G of size < |G|.The main result of this paper reads as follows. If □λ+CHλ holds for a given uncountable cardinal λ, then for every cardinal κ≤λ, there exists an (ℵ0,κ)-chromatic graph of size λ+.We also study (ℵ0,λ+)-chromatic graphs of size λ+. In particular, it is proved that if 0# does not exist, then for every singular strong limit cardinal λ, there exists an (ℵ0,λ+)-chromatic graph of size λ+.

On the consistency strength of the Milner–Sauer conjecture

Abstract In their paper from 1981, Milner and Sauer conjectured that for any poset 〈 P , ≤ 〉 , if cf ( P , ≤ ) = λ > cf ( λ ) = κ , then P must contain an antichain of cardinality κ . The conjecture is consistent and known to follow from GCH-type assumptions. We prove that the conjecture has large cardinals consistency strength in the sense that its negation implies, for example, the existence of a measurable cardinal in an inner model. We also prove that the conjecture follows from Martin’s Maximum and holds for all singular λ above the first strongly compact cardinal.

A topological reflection principle equivalent to Shelah’s strong hypothesis

We notice that Shelahs Strong Hypothesis is equivalent to the following reflection principle: Suppose (X, r) is a first-countable space whose density is a regular cardinal, κ. If every separable subspace of X is of cardinality at most κ, then the cardinality of X is κ.

REDUCED POWERS OF SOUSLIN TREES

We study the relationship between a

Strong failures of higher analogs of Hindman’s theorem

\kappa

The failure of diamond on a reflecting stationary set

-Souslin tree

Hedetniemi's conjecture for uncountable graphs

T

A microscopic approach to Souslin-tree constructions, Part I

and its reduced powers

On guessing generalized clubs at the successors of regulars

T^\theta/\mathcal U

SQUARE WITH BUILT-IN DIAMOND-PLUS

. Previous works addressed this problem from the viewpoint of a single power