##### Department of Mathematics,

University of California San Diego

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### Math 269 - Combinatorics

## Lina Li

#### University of Illinois at Urbana-Champaign

## An overview of Erd\H{o}s--Rothschild problems and their rainbow variants

##### Abstract:

n 1974, Erd\H{o}s and Rothchild conjectured that the complete bipartite graph has the maximum number of two-edge-colorings without monochromatic triangles over all n-vertex graphs. Since then, a new class of colored extremal problems has been extensively studied by many researchers on various discrete structures, such as graphs, hypergraphs, Boolean lattices and sets. In this talk, I will first give an overview of some previous results on this topic. The second half of this talk is to explore the rainbow variants of the Erd\H{o}s-Rothschild problem. With Jozsef Balogh, we confirm conjectures of Benevides, Hoppen and Sampaio, and Hoppen, Lefmann, and Odermann, and complete the characterization of the extremal graphs for the edge-colorings without rainbow triangles. Next, we study a similar question on sum-free sets, where we describe the extremal configurations for integer colorings with forbidden rainbow sums. The latter is joint work with Yangyang Cheng, Yifan Jing, Wenling Zhou and Guanghui Wang.

Host: Jacques Verstraete

### December 3, 2019

### 1:00 PM

### AP&M 7321

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