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Topic

Counting X-free sets

Department of Mathematics and Statistics

Date & location

• Thursday, June 13, 2024
• 2:30 P.M.
• David Strong Building, Room C108

Examining Committee

Supervisory Committee

• Dr. Natasha Morrison, Department of Mathematics and Statistics, ·¬ÇÑÉçÇø (Supervisor)
• Dr. Jonathan Noel, Department of Mathematics and Statistics, UVic (Co-Supervisor)
• Dr. Anthony Quas, Department of Mathematics and Statistics, UVic (Member)

External Examiner

• Dr. Adam Zsolt Wagner, Department of Mathematical Sciences, Worcester Polytechnic Institute

Chair of Oral Examination

• Dr. Panajotis Agathoklis, Department of Electrical and Computer Engineering, UVic

Abstract

Let 𝑋 be a finite subset of ℤ^𝑑. A set 𝐴 ⊆ [𝑛]^𝑑 is 𝑋-free if it does not contain a copy of 𝑋, that is subset of the form 𝒃 + 𝑟 ⋅ 𝑋 for any 𝑟 > 0 and 𝒃 ∈ ℝ^𝑑. Let 𝑟_𝑋(𝑛) denote the cardinality of the largest 𝑋-free subset of [𝑛]^𝑑. In this thesis we explore 𝑋-free sets in three ways. Firstly, we give an exposition of a standard multidimensional extension of Behrend’s construction that gives a lower bound on 𝑟_𝑋(𝑛) for all |𝑋| ≥ 3. Next, using this lower bound on 𝑟_𝑋(𝑛), we lower bound the number of copies of 𝑋 guaranteed in subsets with cardinality larger than 𝑟_𝑋(𝑛), a supersaturation result. Finally, using our supersaturation result, we show that for infinitely many values of 𝑛 the number of 𝑋-free subsets is 2^{𝑂(𝑟_𝑋(𝑛))}. This result is obtained using the powerful hypergraph container method. Further, it generalizes previous work of Balogh, Liu, and Sharifzadeh and Kim.

This thesis includes joint work with Natalie Behague, Joseph Hyde, Natasha Morrison, and Jonathan Noel.