[Paper Review] Ramsey numbers for partially ordered sets

Christian Winter|arXiv (Cornell University)|Jan 1, 2024

Advanced Topology and Set Theory1 citations

TL;DR

This dissertation establishes asymptotically tight bounds on poset Ramsey numbers R(P, Qn) for fixed posets P and growing Boolean lattices Qn, introducing novel probabilistic constructions and blocker-based methods. It proves a dichotomy in asymptotic behavior depending on whether P contains V2 or Λ2, and provides the first linear lower bound (2.02^n) and superlinear upper bound (n² − (1−o(1))n log n) for the diagonal Ramsey number R(Qn, Qn).

ABSTRACT

The main objective of Ramsey theory is to investigate the largest monochromatic substructure guaranteed in any coloring of a given discrete host structure. Examples for such substructures are subgraphs hosted in a complete graph or arithmetic progressions in the natural numbers. In this thesis, we present quantitative Ramsey-type results in the setting of finite sets that are equipped with a partial order, so-called posets. A prominent example of a poset is the Boolean lattice $Q_n$, which consists of all subsets of $\{1,\dots,n\}$, ordered by inclusion. For posets $P$ and $Q$, the poset Ramsey number $R(P,Q)$ is the smallest $N$ such that no matter how the elements of $Q_N$ are colored in blue and red, there is either an induced subposet isomorphic to $P$ in which every element is colored blue, or an induced subposet isomorphic to $Q$ in which every element is colored red. The central focus of this thesis is to investigate $R(P,Q_n)$, where $P$ is fixed and $n$ grows large. Our results contribute to an active area of discrete mathematics, which studies the existence of large homogeneous substructures in host structures with local constraints, introduced for graphs by Erd\H{o}s and Hajnal. We provide an asymptotically tight bound on $R(P,Q_n)$ for $P$ from several classes of posets, and show a dichotomy in the asymptotic behavior of $R(P,Q_n)$, depending on whether $P$ contains a subposet isomorphic to one of two specific posets. A fundamental question in the study of poset Ramsey numbers is to determine the asymptotic behavior of $R(Q_n,Q_n)$ for large $n$. In this dissertation, we present improvements on the known lower and upper bound on $R(Q_n,Q_n)$. Moreover, we explore variations of the poset Ramsey setting, including Erd\H{o}s-Hajnal-type questions when the small forbidden poset has a non-monochromatic color pattern, and so-called weak poset Ramsey numbers, which are concerned with non-induced subposets.

Motivation & Objective

• To determine the asymptotic behavior of R(P, Qn) for fixed posets P and large n.
• To resolve the dichotomy in Ramsey growth depending on whether P contains V2 or Λ2 as subposets.
• To improve the best-known lower and upper bounds for the diagonal Ramsey number R(Qn, Qn).
• To explore variations of the poset Ramsey problem, including weak Ramsey numbers and Erdős-Hajnal-type settings.
• To conjecture that R(Qn, Qn) = O(n^{1+o(1)}) and R(P, Qn) = n + o(n) for all fixed P.

Proposed method

• Extends the Chain Lemma and introduces a new blocker-based method for lower bounds.
• Develops a probabilistic construction using parallel factorial trees to achieve a linear improvement in the lower bound for R(Qn, Qn).
• Refines the Blob Lemma to derive a superlinear upper bound on R(Qn, Qn).
• Applies homomorphism and embedding techniques to characterize posets without V2 or Λ2 subposets.
• Uses Y-chains and framework decompositions to analyze induced subposet structures.
• Introduces weak poset Ramsey numbers and studies non-induced subposets.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of R(P, Qn) as n → ∞ for a fixed poset P?
• RQ2 Does the presence of V2 or Λ2 as a subposet in P cause a sharp transition in the growth rate of R(P, Qn)?
• RQ3 Can the diagonal Ramsey number R(Qn, Qn) be bounded below by a superpolynomial function of n?
• RQ4 Is there a linear improvement over the trivial lower bound 2^n for R(Qn, Qn)?
• RQ5 What is the relationship between the weak poset Ramsey number Rw(Qn, Qn) and the standard R(Qn, Qn)?

Key findings

• For any fixed poset P not containing V2 or Λ2, R(P, Qn) ≤ n + c(P), showing linear growth.
• If P contains V2 or Λ2, then R(P, Qn) ≥ n + n/(15 log n), indicating a superlinear lower bound.
• The first linear improvement on the trivial lower bound for R(Qn, Qn) is established: R(Qn, Qn) ≥ 2.02^n + o(1).
• The first superlinear improvement on the basic upper bound is achieved: R(Qn, Qn) ≤ n² − (1−o(1))n log n.
• A new conjecture is proposed: R(Qn, Qn) = O(n^{1+o(1)}), suggesting subquadratic growth.
• The poset Erdős-Hajnal number ˜R(˙P, Qn) admits a linear improvement over the trivial bound, confirming a related conjecture.