2026-09-18T20:50:17Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/1221662026-06-17T14:43:02Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Larry Guth.
Do, Thao Thi Thu.
Massachusetts Institute of Technology. Department of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2019-09-16T22:33:51Z
2019-09-16T22:33:51Z
2019
2019
https://hdl.handle.net/1721.1/122166
1117775088
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2019
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 63-68).
A (hyper)graph is semi-algebraic if its vertices are points in some Euclidean spaces and the (hyper)edge relation is defined by a finite set of polynomial inequalities. Semi-algebraic (hyper)graphs have been studied extensively in recent years, and many classical results in (hyper)graph theory such as Ramsey's theorem and Szemerédi's regularity lemma can be significantly improved in the semi-algebraic setting. In this dissertation, we discuss three problems in incidence geometry where the bounds for semi-algebraic (hyper)graphs are generally better than the ones for arbitrary (hyper)graphs : (1) what is the maximum number of hyperedges in a hypergraph forbidding some pattern? (2) what is the most compact way to decompose a graph by complete bipartite subgraphs? and (3) what is the maximum number of edges in a graph where no two neighbor sets have a large intersection? As most graphs and hypergraphs arising from problems in discrete geometry are semi-algebraic, our results have applications to discrete geometry. The main tools used in our proofs include some version of polynomial partitioning, a Milnor-Thom-type result from topology and a packing-type result in set system theory.
by Thao Thi Thu Do.
Ph.D.
Ph.D. Massachusetts Institute of Technology, Department of Mathematics
68 pages ;
eng
Massachusetts Institute of Technology
MIT theses are protected by copyright. They may be viewed, downloaded, or printed from this source but further reproduction or distribution in any format is prohibited without written permission.
http://dspace.mit.edu/handle/1721.1/7582
Mathematics.
Semi-algebraic graphs and hypergraphs in incidence geometry
Thesis
application/pdf
2019-09-16T22:33:49Z
Publication
Doctoral
Math
unknown
unknown
<Publication xmlns="https://www.openaire.eu/cerif-profile/1.1/" id="72509203-e8c1-490f-b8d8-72fffdfec3c9">
<Type xmlns="https://www.openaire.eu/cerif-profile/vocab/COAR_Publication_Types">http://purl.org/coar/resource_type/c_1843</Type>
<Language>eng</Language>
<Title>Semi-algebraic graphs and hypergraphs in incidence geometry</Title>
<PublishedIn>
<Publication>
</Publication>
</PublishedIn>
<PublicationDate>2019</PublicationDate>
<Authors>
<Author>
<DisplayName>Do, Thao Thi Thu.</DisplayName>
<Affiliation>
<OrgUnit>
</OrgUnit>
</Affiliation>
</Author>
</Authors>
<Editors>
</Editors>
<Publishers>
<Publisher>
<DisplayName>Massachusetts Institute of Technology</DisplayName>
<OrgUnit />
</Publisher>
</Publishers>
<License>http://dspace.mit.edu/handle/1721.1/7582</License>
<Keyword>Mathematics.</Keyword>
<Abstract>A (hyper)graph is semi-algebraic if its vertices are points in some Euclidean spaces and the (hyper)edge relation is defined by a finite set of polynomial inequalities. Semi-algebraic (hyper)graphs have been studied extensively in recent years, and many classical results in (hyper)graph theory such as Ramsey's theorem and Szemerédi's regularity lemma can be significantly improved in the semi-algebraic setting. In this dissertation, we discuss three problems in incidence geometry where the bounds for semi-algebraic (hyper)graphs are generally better than the ones for arbitrary (hyper)graphs : (1) what is the maximum number of hyperedges in a hypergraph forbidding some pattern? (2) what is the most compact way to decompose a graph by complete bipartite subgraphs? and (3) what is the maximum number of edges in a graph where no two neighbor sets have a large intersection? As most graphs and hypergraphs arising from problems in discrete geometry are semi-algebraic, our results have applications to discrete geometry. The main tools used in our proofs include some version of polynomial partitioning, a Milnor-Thom-type result from topology and a packing-type result in set system theory.</Abstract>
<Access xmlns="http://purl.org/coar/access_right"
>
</Access>
</Publication>