Basis problem for analytic multiple gaps

TL;DR

This paper establishes a finite basis for all analytic multiple gaps of a fixed size, develops their detailed structure, and explores applications, advancing the understanding of combinatorial set theory and infinite subset partitions.

Contribution

It proves the existence of a finite basis for analytic k-gaps and develops their fine structure theory, with several applications and connections to previous publications.

Findings

Existence of a finite basis for all analytic k-gaps.

Development of the fine structure theory of analytic k-gaps.

Multiple applications in the theory of infinite subsets and partitions.

Abstract

A k-gap is a finite k-sequence of pairwise disjoint monotone families of infinite subsets of N mixed in such a way that we cannot find a partition of N such that each family is trival on one piece of the partition. We prove that, relative to the comparison given by restriction to infinite subsets of N, for every positive integer k there is a finite basis for the class of all analytic k-gaps . We also build the fine structure theory of analytic k-gaps and give some applications. The content of Chapter 1 of this manuscript have been published as: A. Avil\'es, S. Todorcevic, Finite basis for analytic multiple gaps, Publ. Math. IHES. 121 (2015), 57-79. The content of Chapter 2 (except some technical results from 2.5 and 2.6) and Section 3.1, largely revised and improved, has ben published as: A. Avil\'es, S. Todorcevic, Types in the n-adic tree and minimal analytic gaps, Adv. Math. 292 (2016), 558-600. The content of Sections 3.4, 4.1 and 4.3 have been published as: A. Avil\'es, S. Todorcevic, Isolating subgaps of a multiple gap, Monatsh. Math. 186 (2018), 373--392. The rest of contents may appear elsewhere.