Date of Award

6-15-2024

Document Type

Dissertation

Degree Name

Ph.D.

Organizational Unit

College of Natural Science and Mathematics, Mathematics

First Advisor

Paul Horn

Second Advisor

Mei Yin

Third Advisor

Chris GauthierDickey

Fourth Advisor

Shashank Kanade

Fifth Advisor

Petr Vojtechovsky

Keywords

Colorings, Enumeration, Extremal problems, Games, Integers, Permutations

Abstract

This dissertation consists of several combinatorial problems on the integers. These problems fit inside the areas of extremal combinatorics and enumerative combinatorics.

We first study monochromatic solutions to equations when integers are colored with finitely many colors in Chapter 2. By looking at subsets of {1, 2, . . . , n} whose least common multiple is small, we improved a result of Brown and Rödl on the smallest integer n such that every 2-coloring of {1, 2, . . . , n} has a monochromatic solution to equations with unit fractions. Using a recent result of Boza, Marín, Revuelta, and Sanz, this technique also allows us to show a polynomial upper bound for the same problem, but with three colors.

We then study Maker-Breaker positional games for equations with fractional powers in Chapter 3. In these games, Maker and Breaker take turns to select a previously unclaimed number in {1, 2, . . . , n}, Maker wins if they can form a solution to a given equation, and Breaker wins if they can stop Maker. Using combinatorial arguments and results from number theory and arithmetic Ramsey theory, we found exact expressions or strong bounds for the smallest n such that Maker has a winning strategy.

Finally, we study permutations of integers in Chapters 4 to 6. In Chapter 4, we provide an alternative proof of a result by Miner and Pak which says that 123- and 132-avoiding permutations with a fixed leading term are enumerated by the ballot numbers. We then study the number of pattern-avoiding permutations with a fixed prefix of length t ≥ 1, generalizing the t = 1 case. We find exact expressions for single and pairs of patterns of length three as well as the pair 3412 and 3421. These expressions depend on t, the extrema, and the order statistics. In Chapter 5, we define rotations of permutations and study permutations such that they and their rotations avoid certain patterns. We obtain many enumerative results for patterns of length three and several of them are related to existing results on permutations avoiding other patterns. In Chapter 6, we look at subsequences with certain arithemtic properties that exist in all permutations of a given length. For example, we prove that for all positive integers k ≥ 3 and sufficiently large n, every permutation of {1, 2, . . . , n} has a subsequence (a1, a2, . . . , ak) such that either ∑ki=1 ai = 2a1 or ∑ki=1 ai = 2ak.

Copyright Date

6-2024

Copyright Statement / License for Reuse

All Rights Reserved
All Rights Reserved.

Publication Statement

Copyright is held by the author. User is responsible for all copyright compliance.

Rights Holder

Collier Gaiser

Provenance

Received from ProQuest

File Format

application/pdf

Language

English (eng)

Extent

160 pgs

File Size

705 KB

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