6. 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

10. 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

16. This graph has a girth of 6

24. http://en.wikipedia.org/wiki/Lov%C3%A1sz_loca l_lemma http://en.wikipedia.org/wiki/Lov%C3%A1sz_loca l_lemma http://en.wikipedia.org/wiki/Tournament_(graph_t heory) http://en.wikipedia.org/wiki/Tournament_(graph_t heory http://en.wikipedia.org/wiki/Hamiltonian_path http://en.wikipedia.org/wiki/Hypergraph http://en.wikipedia.org/wiki/Ramsey%27s_theore m http://en.wikipedia.org/wiki/Ramsey%27s_theore m http://en.wikipedia.org/wiki/Girth_(graph_theory) http://en.wikipedia.org/wiki/Girth_(graph_theory http://mathworld.wolfram.com/ChromaticNumber.html http://mathworld.wolfram.com/ChromaticNumber.html http://statwiki.ucdavis.edu/Under_Construction/D escriptive_Statistics/2.5_The_Empirical_Rule_an d_Chebyshev's_Theorem