Counting X-free sets
Date
2024
Authors
Wright, Ashna Keaton
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Abstract
Let X be a finite subset of the integers. A set A contained in [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 in the d-dimensional real numbers. 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, we lower bound the number of copies of X guaranteed in subsets with cardinality larger than r_X(n). This is a supersatuation}result that uses the previously demonstrated lower bound on r_X(n). 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 A. Noel.
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Keywords
Extremal combinatorics, Combinatorics, Supersaturation, Hypergraph container method