Disproving the Neighborhood Conjecture

We study the following Maker/Breaker game. Maker and Breaker take turns in choosing vertices from a given n-uniform hypergraph F, with Maker going first. Maker’s goal is to completely occupy a hyperedge and Breaker tries to avoid this. Beck conjectures that if the maximum neighborhood size of F is at most 2^(n-1) then Breaker has a winning strategy. We disprove this conjecture by establishing an n-uniform hypergraph with maximum neighborhood size 3*2^(n-3) where Maker has a winning strategy. Moreover, we show how to construct an n-uniform hypergraph with maximum degree 2^(n-1)/n where Maker has a winning strategy. Finally we show that each n-uniform hypergraph with maximum degree at most 2^(n-2)/(en) has a proper halving 2-coloring, which solves another open problem posed by Beck related to the Neighborhood Conjecture.

💡 Research Summary

The paper investigates a positional game known as the Maker/Breaker game on n‑uniform hypergraphs. In this game two players alternately claim previously unclaimed vertices; Maker moves first and wins if he eventually occupies all vertices of some hyperedge, while Breaker wins if he can prevent this. A classical result, the Erdős–Selfridge theorem, states that any n‑uniform hypergraph with fewer than 2ⁿ − 1 hyperedges is a Breaker win. Beck conjectured a stronger “Neighbourhood Conjecture”: if the maximum neighbourhood size (the number of hyperedges intersecting a given hyperedge) is smaller than 2ⁿ⁻¹, then Breaker should have a winning strategy.

The authors disprove this conjecture by constructing explicit counter‑examples. Their constructions belong to a class C of hypergraphs where every hyperedge corresponds to a root‑to‑leaf path in a binary tree T_H. They first fix a deterministic winning strategy for Maker: start at the root and, at each turn, move to a child whose subtree does not contain Breaker’s most recent vertex. Because Breaker can occupy at most one vertex per level, such a child always exists, guaranteeing that Maker can claim a full root‑to‑leaf path, i.e., a hyperedge. Observation 2.1 formalises that any hypergraph in C where every full branch contains at least one hyperedge is a Maker win.

To obtain a hypergraph with relatively small neighbourhood size yet still a Maker win, they recursively augment the binary tree. Starting from a leaf, they attach two children; one child immediately forms a hyperedge (the path from the leaf to that child), while the other child becomes the root of a subtree of height n − 2. This process is repeated on every leaf of the new subtree, ensuring that every full branch eventually contains a hyperedge. Careful counting shows that each hyperedge intersects at most 2ⁿ⁻² + 2ⁿ⁻³ other hyperedges, so the maximum neighbourhood size of the resulting hypergraph H is 3·2ⁿ⁻³, which is strictly below Beck’s bound 2ⁿ⁻¹. Thus Beck’s conjecture is false.

The second main result concerns the maximum vertex degree rather than neighbourhood size. The authors build an n‑uniform hypergraph with maximum degree 2ⁿ⁻¹ / n on which Maker still wins. The construction proceeds in several stages. First they create a non‑uniform hypergraph with maximum degree 2d, where d = 2ⁿ / n, by assigning to each node v of a binary tree a collection of hyperedges that are the paths from v to each leaf in its subtree, each duplicated 2^{level(v)} times. This yields a degree bound of 2d. Next they introduce the notion of a “unit”: a set of 2ᶦ hyperedges of size log d + 1 − i for some i. By attaching auxiliary binary subtrees of logarithmic height to leaves and augmenting existing hyperedges with vertices from these subtrees, they transform the non‑uniform structure into an n‑uniform one while preserving the degree bound. After a careful hierarchy of augmentations (including a second logarithmic layer and finally a layer that expands each unit to a single n‑vertex hyperedge), they obtain a hypergraph H with maximum degree exactly 2ⁿ⁻¹ / n and the property that every full branch of its underlying tree contains a hyperedge. Consequently Maker has a forced win, disproving a stronger version of Beck’s conjecture that would replace the neighbourhood bound by a degree bound of the same order.

Finally, the authors address a related open problem posed by Beck concerning “proper halving 2‑colorings”. They prove that any n‑uniform hypergraph whose maximum degree does not exceed 2ⁿ⁻² / (e n) admits a 2‑coloring in which the numbers of red and blue vertices differ by at most one, and every hyperedge contains vertices of both colors. The proof is an application of the Lovász Local Lemma (specifically its algorithmic version), showing that the probability of a bad event (a hyperedge being monochromatic) can be made sufficiently small when the degree condition holds. This halving coloring guarantees that after the game ends, Breaker will own at least one vertex in each hyperedge, ensuring his victory. Thus the paper resolves Beck’s question about the existence of such colorings under a near‑optimal degree condition.

In summary, the paper makes three substantial contributions: (1) it provides explicit counter‑examples to Beck’s Neighbourhood Conjecture with maximum neighbourhood size 3·2ⁿ⁻³, (2) it constructs n‑uniform hypergraphs of maximum degree 2ⁿ⁻¹ / n that are Maker wins, refuting stronger degree‑based variants of the conjecture, and (3) it establishes that hypergraphs with maximum degree ≤ 2ⁿ⁻² / (e n) always admit proper halving 2‑colorings, thereby solving another open problem of Beck. These results deepen the understanding of positional games on hypergraphs and illustrate the delicate interplay between degree, neighbourhood size, and coloring properties.