A resolution of the Aharoni-Korman conjecture

TL;DR

This work resolves the Aharoni–Korman fishbone conjecture in two senses: it provides a counterexample showing the conjecture fails in full generality for FAC posets, yet proves the conjecture for a broad, natural class of countable posets called vacillating FAC posets that avoid certain infinite-interval configurations. Central to the positive results is the development of chain-extension techniques (H(P)), reduction and bicomparability notions, and the construction of strongly maximal and thick chains, which together yield spines in the vacillating setting. The paper also investigates the existence and structure of strongly maximal chains, and presents a spineless counterexample $P_5$ proving that spine-fulfillment is not guaranteed in general. Overall, the results provide a near-complete resolution by delineating a large tractable domain where AK holds, while clearly separating it from the pathological cases where it fails, thus advancing understanding of infinite poset decomposition and spine structures.

Abstract

A poset $P$ is said to satisfy the finite antichain condition, or FAC for short, if it has no infinite antichain. It was conjectured by Aharoni and Korman in 1992 that any FAC poset $P$ possesses a chain $C$ and a partition into antichains such that $C$ meets every antichain of the partition. Our main results are twofold. We provide a counterexample to the conjecture in full generality, but, despite this, we also prove that the conjecture does hold true for a broad class of posets. In particular, we prove that the Aharoni-Korman conjecture holds for countable posets avoiding intervals $I$ such that either $I$ or its reverse $I^*$ is of the form $\bigoplus_{x\inω} Q_x$, where each $Q_x$ is infinite and co-wellfounded. In pursuit of these goals, we also investigate other facets of the structure of FAC posets. In particular, we consider strongly maximal chains in FAC posets, proving some results, and posing several questions and conjectures.

Figures (4)

• Figure 1: A Hasse diagram of $P_1$. The regions in grey triangles are order-isomorphic to $\omega$.
• Figure 2: A Hasse diagram of $P_2$. The regions in grey triangles are order-isomorphic to $\omega$, and this structure continues infinitely in both directions.
• Figure 3: An approximate Hasse diagram of $P_5$. The regions within the grey V-shapes are order-isomorphic to $\omega\times\omega$, and this structure continues infinitely downwards. Relations between $L_n$ and $L_{n+1}$ are shown in more detail in \ref{['fig:counterexample-details']}.
• Figure 4: Two examples of the relations between $L_n$ and $L_{n+1}$. The red region in $L_n$ is the set of those points above $(1,10,n+1)$ (shown in a red circle), and the blue region in $L_{n+1}$ is the set of those points below $(1,2,n)$ (shown in a blue circle).

Theorems & Definitions (116)

• Conjecture 1.1: aharoni1992greene, Conjecture 4.1
• Theorem 1.2
• Theorem 1.3
• Definition 1.4
• Definition 1.5
• Theorem 1.8: aharoni1992greene, Theorem 3.1
• Conjecture 1.9: aharoni1992greene, Conjecture 4.1
• Theorem 1.10: duffus2002intervals, Theorem 4.1
• Theorem 1.11: zaguia2024progress, Theorem 6
• Definition 1.12