# A Collection of Results of Simonyi's Conjecture

 Title: A Collection of Results of Simonyi's Conjecture Styner, Dustin Department of Mathematics and Statistics Mathematics and Statistics Pereira, RajeshMcNicholas, Paul Given two set systems $\mathscr{A}$ and $\mathscr{B}$ over an $n$-element set, we say that $(\mathscr{A,B})$ forms a recovering pair if the following conditions hold: \\ $\forall A, A' \in \mathscr{A}$ and $\forall B, B' \in \mathscr{B}$, $A \setminus B = A' \setminus B' \Rightarrow A=A'$ \\ $\forall A, A' \in \mathscr{A}$ and $\forall B, B' \in \mathscr {B}$, $B \setminus A = B' \setminus A' \Rightarrow B=B'$ \\ In 1989, G\'{a}bor Simonyi conjectured that if $(\mathscr{A,B})$ forms a recovering pair, then $|\mathscr{A}\mathscr{B}|\leq 2^n$. This conjecture is the focus of this thesis. This thesis contains a collection of proofs of special cases that together form a complete proof that the conjecture holds for all values of $n$ up to 8. Many of these special cases also verify the conjecture for certain recovering pairs when $n>8$. We also present a result describing the nature of the set of numbers over which the conjecture in fact holds. Lastly, we present a new problem in graph theory, and discuss a few cases of this problem. http://hdl.handle.net/10214/4926 2012-12 All items in the Atrium are protected by copyright with all rights reserved unless otherwise indicated.
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