Abstract
Anti-Ramsey theory concerns itself with finding the fewest number of classes into which a large set must be partitioned, so that no small subset is completely partitioned. Typically, this is studied by considering the rainbow number of $\mathcal F$ with respect to $H$, denoted $\rb(H,\mathcal F)$, which is the smallest integer $r$ such that any $r$-coloring of the host object $H$ admits a rainbow sub-object in the family $\mathcal F$. This thesis studies $\rb(H, \mathcal F)$ in three settings: where the host object is a simple graph, a tournament, or a cyclic group. In particular, Chapter \ref{RainbowMatchings} focuses on the conditions on colorings of different graph families that force rainbow matchings of size $2m$ for some parameter $m$. Chapters \ref{schurpaper} and \ref{SidonPaper} consider the rainbow number for solution sets to equations in cyclic groups. Chapter \ref{tournaments} takes the host object to be tournaments on $n$ vertices, and determines the rainbow number for some families of directed graphs.
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