Growth of regular partitions 4: strong regularity and the pairs partition · arXivDesk
2404.02030Apr 2, 2024Updated to make explicit the fact that all proofs require only a polynomial growth rate for the function measuring the regularity of the pairs partition. Substantial details added to the upper bounds section and corrections of some errors. Contains overlap in preliminaries and background with its companion paper arXiv: 2404.02024
Growth of regular partitions 4: strong regularity and the pairs partition
This paper studies bounds in a strong form of regularity for 3-uniform hypergraphs which was developed by Frankl, Gowers, Kohayakawa, Nagle, Rödl, Skokan, and Schacht. Regular decompositions of this type involve two structural components: a partition on the vertex set and a partition on the pairs of vertices. The regularity of such decompositions are measured by two parameters: an ε1>0 and a function ε2:N→(0,1]
Nearby in the stack
. To each hereditary property
H
of
3
-uniform hypergraphs, we associate two corresponding growth functions:
TH(ε1,ε2)
for the size of the vertex component, and
LH(ε1,ε2)
for the size of the pairs component. The problem of understanding the asymptotic growth of such functions was introduced in a companion paper, which also proved several results about
TH
. In this paper we study the possible asymptotic behavior of
LH
. We show any such function is either constant, bounded above and below by a polynomial, or bounded below by an exponential. All results require only reasonable growth rates for