Best illustrated through many methods Ramsey Numbers Any sufficiently large graph contains either a clique (a set of vertices that induce a subgraph) or an independent set (a set of vertices that induce edgeless subgraph) Hypergraph Coloring A hypergraph is c-colorable if the vertices of the graph can be colored with c colors given that at least two colors appear in every edge Erdos-Ko-Rado Theorem A family of sets F is intersecting if for every pair of sets within F, the intersection set of both those sets is greater than zero Overall goal: prove a lower bound exists for each method when dealing with large integer values More sophisticated and simpler than counting arguments

Assume the set Ai is an element of F Therefore any other set in As is one of the following: Ai, Ai+1,..., Ai+k-1 These 2k-2 remaining sets can be divided into k-1 pairs (As, As+k) Because n ≥ 2k, the intersection between As and As+k is 0, and therefore only one set from each of the k

This graph has a girth of 6

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