On a problem of Erdos and Hajnal

TL;DR

The paper addresses Erdos–Hajnal-type partition relations at successors of strong limit singular cardinals, proving that one can force $\lambda^+\nrightarrow(\lambda^+,(3)_{\mathrm{cf}(\lambda)})^2$ with $2^\lambda>\lambda^+$ without GCH, and pushing the result down to $\aleph_\omega$. It develops two main methods: a pcf-based lifting from below $\lambda$ to $\lambda^+$ that yields a coloring witnessing the negative arrow, and a stick/tiltan-based approach deriving the same failure from a prediction principle ${\bullet}(\lambda)$. The results are substantiated by forcing constructions starting from large cardinals (e.g., supercompact, extender-based Prikry) to realize the hypotheses, including a path to $\lambda=\aleph_{\omega^2}$ and a refined $\aleph_\omega$ case via a filtered pcf framework. The work also discusses the consistency strength and open questions about whether the negative arrow can hold in ZFC or under weaker assumptions, and contributes to the Erdős–Hajnal program by showing consistency of negative arrow at successors of strong limit singular cardinals without assuming $2^\lambda=\lambda^+$.

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 $\aleph_ω$.

Theorems & Definitions (15)

• Theorem 1
• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Claim 2.1
• Theorem 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Claim 2.6