## Graduation Year

2019

## Document Type

Dissertation

## Degree

Ph.D.

## Degree Name

Doctor of Philosophy (Ph.D.)

## Degree Granting Department

Mathematics and Statistics

## Major Professor

Brendan Nagle, Ph.D.

## Committee Member

Jay Ligatti, Ph.D.

## Committee Member

Brian Curtin, Ph.D.

## Committee Member

Natǎsa Jonoska, Ph.D.

## Committee Member

Theodore Molla, Ph.D.

## Committee Member

Dmytro Savchuk, Ph.D.

## Keywords

Density, Extremal Combinatorics, Links, Partition, Quasirandomness

## Abstract

*Szemere´di’s** **Regularity** **Lemma** *[32, 33] is an important tool in combinatorics, with numerous appli- cations in combinatorial number theory, discrete geometry, extremal graph theory, and theoretical computer science.

The Regularity Lemma hinges on the following concepts. Let *G *= (*V,** E*) be a graph and let ∅ /= *X, Y *⊂ *V *be a pair of disjoint vertex subsets. We define the *density *of the pair (*X, Y *) by *d _{G}*(

*X,*

*Y*) = |

*E*[

*X,*

*Y*]|

*/*(|

*X*||

*Y*|) where

*E*[

*X,*

*Y*] denotes the set of edges {

*x, y*} ∈

*E*with

*x*∈

*X*and

*y*∈

*Y*. We say the pair (

*X, Y*) is

*ε-regular*if all subsets

*X*

^{I}⊆

*X*and

*Y*

^{I}⊆

*Y*satisfying |

*X*

^{I}|

*> ε*|

*X*| and |

*Y*

^{I}|

*> ε*|

*Y*| also satisfy |

*d*(

_{G}*X*

^{I}

*, Y*

^{I}) −

*d*(

_{G}*X, Y*)|

*< ε*.

The Regularity Lemma states that, for all *ε **> *0, all large *n*-vertex graphs *G *= (*V,** **E*) admit a partition *V *= *V*_{1} ∪ · · · ∪ *V _{t}*, where

*t*=

*t*(

*ε*) depends on

*ε*but not on

*n*, so that all but

*εt*

^{2}pairs (

*V*

_{i}*,*

*V*

*), 1 ≤*

_{j}*i < j*≤

*t*, are

*ε*-regular. While Szemere´di’s original proof demonstrates the

*existence*of such a partition, it gave no method for (efficiently)

*constructing*such partitions. Alon, Duke, Lefmann, Ro¨dl, and Yuster [1, 2] showed that such partitions can be constructed in time

*O*(

*M*(

*n*)), where

*M*(

*n*) is the time needed to multiply two

*n*×

*n*{0

*,*1}-matrices over the integers. Kohayakawa, Ro¨dl, and Thoma [17, 18] improved this time to

*O*(

*n*

^{2}).

The Regularity Lemma can be extended to *k*-uniform hypergraphs, as can algorithmic for- mulations thereof. The most straightforward of these extends the concepts above to *k*-uniform hypergraphs H = (*V, **E*) in a nearly verbatim way. Let ∅ /= *X*_{1}*, . . . , X _{k} *⊂

*V*be pairwise disjoint subsets, and let

*E*[

*X*

_{1}

*,*

*. . . , X*] denote the set of

_{k}*k*-tuples {

*x*

_{1}

*,*

*. . . , x*} ∈

_{k}*E*satisfying

*x*

_{1}∈

*X*

_{1}

*,*

*. . . , x*∈

_{k}*X*. We define the

_{k}*density*of (

*X*

_{1}

*,*

*. . . , X*) as

_{k}*d*_{H}(*X*_{1}*, . . . , X _{k}*) = |

*E*[

*X*

_{1}

*, . . . , X*]| / |

_{k}*X*

_{1}| · · · |

*X*|.

_{k}We say that (*X*_{1}*, **. . . , X _{k}*) is

*ε-regular*if all subsets

*X*

*i*I ⊆

*X*, 1 ≤

_{i}*i*≤

*k*, satisfying |

*X*

*i*I|

*> ε*|

*X*| also satisfy

_{i}|*d*_{H}* *(*X*1I *,** **.** **.** **.** **,** **X**k*I ) − *d _{H} *(

*X*

_{1}

*, . . . , X*)|

_{k}*< ε.*

With these concepts, Szemeredi’s original proof can be applied to give that, for all integers *k *≥ 2 and for all *ε > *0, all *n*-vertex *k*-uniform hypergraphs H = (*V, **E*) admit a partition *V *= *V*_{1} ∪· · ∪ *V _{t}*, where

*t*=

*t*(

*k, ε*) is independent of

*n*, so that all but

*εt*many

^{k}*k*-tuples (

*V*1

_{i}*, . . . , V*

_{i}*k*) are

*ε*-regular, where 1 ≤

*i*

_{1}

*<*· · ·

*< i*≤

_{k}*t*. Czygrinow and Ro¨dl [4] gave an algorithm for such a regularity lemma, which in the context above, runs in time

*O*(

*n*

^{2k−1}log

^{5}

*n*).

In this dissertation, we consider regularity lemmas for 3-uniform hypergraphs. In this setting, our first main result improves the algorithm of Czygrinow and Ro¨dl to run in time *O*(*n*^{3}), which is optimal in its order of magnitude. Our second main result shows that this algorithm gives a stronger notion of regularity than what is described above, where this stronger notion is described in the course of this dissertation. Finally, we discuss some ongoing applications of our constructive regularity lemmas to some classical algorithmic hypergraph problems.

## Scholar Commons Citation

Theado, John, "An Optimal Medium-Strength Regularity Algorithm for 3-uniform Hypergraphs" (2019). *USF Tampa Graduate Theses and Dissertations.*

https://digitalcommons.usf.edu/etd/7969