## Number Theory - Sonja Ruzic

### Tuesday, November 27th, 2018

**Time:** 10:00-11:00 a.m. **Place:** Jeffery Hall 422

**Speaker:** Sonja Ruzic

**Title:** Presentation of the paper "Maxima for Graphs and a New Proof of a Theorem of Turan", by T.S. Motzkin and E.G. Straus.

**Abstract:** In this talk we consider the following problem: Given a graph G with vertices 1, 2, ..., n, let S be the simplex in $\mathbb{R}^n$ given by the set $x={x_1, x_2, ..., x_n | \sum_{i=1}^{n}x_i=1, x_i \geq 0 \forall I}$. What is $\max_{x \in S} \sum_{(i, j)\in G}x_ix_j?$ Furthermore, a proof of a theorem of Turan, which gives an upper bound to the number of edges of a graph G which contains no complete subgraph of order k, will be presented.