On a problem of Erdos and Hajnal

TL;DR

This paper demonstrates that it is possible to construct graphs with no large independent sets and no monochromatic triples as successors of strong limit singular cardinals without assuming GCH, extending previous results.

Contribution

It shows the existence of such graphs at leph_, removing the need for GCH assumptions, thus advancing understanding in set theory and graph theory.

Findings

Graphs with no large independent sets and no monochromatic triples exist at leph_.

The construction does not rely on GCH assumptions.

Results extend to leph_, including leph_.

Abstract

Addressing a question of Erdos and Hajnal we show that one can force a graph with no large independent sets and no monochromatic triples as successors of strong limit singular cardinals, even without assuming GCH. The result can be pushed down to $\aleph_\omega$.