Growth of regular partitions 3: strong regularity and the vertex partition · arXivDesk
2404.02024Apr 2, 2024Updated to make explicit which results require polynomial vs. super polynomial growth of the function determining the regularity of the pairs partition. Contains some overlap of preliminaries and background material with its companion paper arXiv:2404.02030
Growth of regular partitions 3: strong regularity and the vertex partition
We consider here the strong regularity for 3-uniform hypergraphs developed by Frankl, Gowers, Kohayakawa, Nagle, Rödl, Skokan, and Schacht. This type of regular decomposition comes with two components, a partition of the vertices, and a partition of the pairs of vertices. The data of a regular decomposition also includes two parameters measuring quasirandomness, a fixed constant ε1>0, and a function ε2:N→(0,1]
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. We define two growth functions associated to a hereditary property
H
of
3
-uniform hypergraphs:
TH(ε1,ε2)
which measures the size of the vertex component, and
LH(ε,ε2)
which measures the size of the pairs component. We introduce the following question. What are the possible asymptotic growth rates of functions of the form
TH
and
LH
? In this paper, we consider this question for
TH
, proving a separation into four classes: constant, polynomial, exponential, or at least wowzer. The separations among the constant, polynomial and exponential ranges require only slow growing (namely polynomial) choices for
ε2
. The jump to the wowzer range uses a very fast growing
ε2
and makes crucial use of a lower bound construction for strong graph regularity due to Conlon and Fox.