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What is the maximum number of edges in a graph on *n* vertices without triangles? Mantel’s answer in 1907—that at most half of the edges can be present—started a new field: extremal combinatorics. More generally, what is the maximum number of edges in an *n*-vertex graph that does not contain any subgraph isomorphic to H? What about if you consider hypergraphs instead of graphs? I will introduce the technique of sums of squares and discuss how it can be used to attack such problems.

#### Contact Info

Tracie Rubeck

(413) 542-2100

@amherst.edu