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Topic

Quasirandom forcing in Regular Tournaments

Department of Mathematics and Statistics

Date & location

• Friday, April 11, 2025
• 10:00 A.M.
• Clearihue Building, Room B021

Examining Committee

Supervisory Committee

• Dr. Jonathan Noel, Department of Mathematics and Statistics, ßÉßɱ¬ÁÏ (Supervisor)
• Dr. Jane Butterfield, Department of Mathematics and Statistics, UVic (Co-Supervisor)

External Examiner

• Dr. Leonardo Coregliano, Department of Mathematics, University of Chicago

Chair of Oral Examination

• Dr. Clifford Roberts, Department of Philosophy, UVic

Abstract

The study of quasirandom forcing in various discrete structures has been a wellk-nown problem in Extremal Combinatorics since 1987. In this work, we study quasirandom forcing in the case of tournaments. A tournament 𝐻 forces quasirandomness if it has the property that every sequence (𝑇_𝑛)_𝑛∈ℕ of tournaments of increasing order is quasirandom if and only if the density of 𝐻 in 𝑇_𝑛 asymptotically equals its expected value as 𝑛→∞. In contrast to the analogous problem in graphs, it was shown that there exists only one non-transitive tournament that forces quasirandomness. To obtain a richer family of tournaments with this property, we propose a variant of it restricting the definition of quasirandom forcing to only nearly regular sequences of tournaments (𝑇_𝑛)_𝑛∈ℕ. We characterize all tournaments on at most 5 vertices that forces quasirandomness under this new setting, obtaining that 11 out of 16 tournaments on at least four vertices are quasirandom forcing.