Infinite Games and Ramsey Properties of $F_\sigma$ Ideals

TL;DR

This paper explores combinatorial and Ramsey properties of Borel ideals on countable sets, extending known theorems, identifying critical ideals for Ramsey properties with multiple colors, and addressing open questions in the field.

Contribution

It extends a theorem on Ramsey properties of ideals, identifies an $F_\sigma$ tall ideal with a winning strategy in the Cut and Choose Game, and analyzes the Ramsey properties of the random graph ideal for multiple colors.

Findings

An $F_\sigma$ tall ideal with a winning strategy for player II in the Cut and Choose Game.

The random graph ideal is critical for Ramsey properties with more than two colors.

Identification of an $F_\sigma$ tall $K$-uniform ideal not equivalent to $ ext{ED}_{ ext{fin}}$.

Abstract

In this work, we investigate various combinatorial properties of Borel ideals on countable sets. We extend a theorem presented in M. Hru\v{s}\'{a}k, D. Meza-Alc\'antara, E. Th\"ummel, and C. Uzc\'ategui, \emph{Ramsey Type Properties of Ideals}, and identify an $F_\sigma$ tall ideal in which player II has a winning strategy in the Cut and Choose Game, thereby addressing a question posed by J. Zapletal. Additionally, we explore the Ramsey properties of ideals, demonstrating that the random graph ideal is critical for the Ramsey property when considering more than two colors. The previously known result for two colors is extended to any finite number of colors. Furthermore, we comment on the Solecki ideal and identify an $F_\sigma$ tall $K$-uniform ideal that is not equivalent to $\mathcal{ED}_{\text{fin}}$, thereby addressing a question from Michael Hru\v{s}\'ak.