Is it easy to regularize a hypergraph with easy links?
Lior Gishboliner, Asaf Shapira, Yuval Wigderson
TL;DR
This work resolves a central question in hypergraph regularity by showing that, for k-graphs whose vertex links each admit a polynomial-sized ε′-homogeneous partition, there is a global ε-homogeneous equipartition of size 2^{(r/ε)^{O(1)}} (single exponential). It also establishes a matching lower bound: even when every link has a small regular partition, the global weakly regular partition of the whole hypergraph can require tower-type size, answering a longstanding question about the utility of link-level regular partitions. Additionally, a positive result shows that if every link has a small homogeneous partition, then the whole hypergraph inherits a small homogeneous partition (exponential in the link-size bound), bridging VC-dimension concepts with partition regularity. Collectively, the results delineate when local regularity information yields compact global partitions and illuminate the contrasting behavior between regular and homogeneous partitions in hypergraphs, with implications for bounded slicewise VC-dimension and induced-embedding phenomena.
Abstract
A partition of a (hyper)graph is $\varepsilon$-homogenous if the edge densities between almost all clusters are either at most $\varepsilon$ or at least $1-\varepsilon$. Suppose a $3$-graph has the property that the link of every vertex has an $\varepsilon$-homogenous partition of size $\text{poly}(1/\varepsilon)$. Does this guarantee that the $3$-graph also has a small homogenous partition? Terry and Wolf proved that such a $3$-graph has an $\varepsilon$-homogenous partition of size given by a wowzer-type function. Terry recently improved this to a double exponential bound, and conjectured that this bound is tight. Our first result in this paper disproves this conjecture by giving an improved (single) exponential bound, which is best possible. We further obtain an analogous result for $k$-graphs of all uniformities $k \geq 3$. The above problem is part of a much broader programme which seeks to understand the conditions under which a (hyper)graph has small $\varepsilon$-regular partitions. While this problem is fairly well understood for graphs, the situation is (as always) much more involved already for $3$-graphs. For example, it is natural to ask if one can strengthen our first result by only requiring each link to have $\varepsilon$-regular partitions of size $\text{poly}(1/\varepsilon)$. Our second result shows that surprisingly the answer is `no', namely, a $3$-graph might only have regular partitions of tower-type size, even though the link of every vertex has an $\varepsilon$-regular partition of polynomial size.