bibkey: bubbolonicaceres2026neighbourhood authors: Daniela Bubboloni; José Cáceres year: 2026 title: “The neighbourhood convexity” doi: 10.48550/arXiv.2608.25912 url: https://arxiv.org/html/2608.25912v1 claim: “For a finite nonempty simple undirected graph, the paper defines neighbourhood convexity and conjectures that every upset of the neighbourhood preorder is neighbourhood-convex whenever the neighbourhood convexity is a convex geometry. Lemma 29(iii) already proves this for principal upsets; the conjecture concerns all upsets.” strata_touched:

• D5/S3/Combinatorics/Graph/BubboloniCaceresNeighbourhoodUpsets license: citation-only triage: anchor

The neighbourhood convexity

Verified locator

Daniela Bubboloni and José Cáceres, arXiv:2608.25912v1, version published 2026-08-26. DOI: 10.48550/arXiv.2608.25912. The primary versioned HTML is https://arxiv.org/html/2608.25912v1. Its inspected SHA256 is f10465fb6352f039eecb0bebd801fd4b839c50b7154c1c36e91a1409632e24af. The verified scope is Sections 2.1 and 2.2, Definitions 3, 5, 9 and 28, Proposition 12, Lemma 29, and the exact unnumbered upset conjecture in “Conclusions and future lines of work” quoted below. This note quotes and cites the source; it does not redistribute the paper.

Source scope and conjecture

Sections 2.1 and 2.2 fix a finite nonempty simple undirected graph and the convex-geometry condition. Definitions 3, 5 and 9 and Proposition 12 give

• $B(x)=\{x\}\cup \{y:xy\in E\}$,
• $N(X)={\bigcap }_{x\in X}B(x)$, with $N(\varnothing )=V$,
• $C=\{N(Y):Y\subseteq V\}\cup \{\varnothing \}$,
• $h(\varnothing )=\varnothing$ and $h(X)=N(N(X))$ for $X\ne \varnothing$,
• $\mathrm{ex}(K)=\{x\in K:K\setminus \{x\}\in C\}$.

The full geometry hypothesis is $h(\mathrm{ex}(K))=K$ for every $K\in C$. Definition 28 orders vertices by $B(x)\subseteq B(y)$. Lemma 29(iii) identifies the principal upset of $x$ with $N(N(\{x\}))$ and proves that it is convex. Lemma 29(iv) proves that convex sets are upsets. Neither is a proof of arbitrary union closure.

In “Conclusions and future lines of work,” in the paragraph beginning “By exploring the connection with the neighbourhood preorder P(G),” the explicit unnumbered conjecture is:

We conjecture that, given a convex geometry (G,n), any upset in P(G) is an n-convex set, having checked it computationally for graphs up to 9 vertices.

The finite computations are author-reported and are not independently verified here. The separate conjecture about extreme points is not this assertion. The source’s statement remains a conjecture in the cited version; the project result is a separate proof, not a published settlement by the original authors.