##### Department of Mathematics,

University of California San Diego

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### Graduate Student Combinatorics Seminar

## Jason O'Neill

#### UCSD

## The Kruskal-Katona Theorem

##### Abstract:

Given an $r$-uniform hypergraph $\mathcal{A} \subset X^{(r)}$, the (lower) shadow of $\mathcal{A}$, denoted $\delta(\mathcal{A})$ is defined as $\delta(\mathcal{A}):= \{ B \in X^{(r-1)} : B \subset A \text{ for some } A \in \mathcal{A} \}$. In this talk, we will explore the classical Kruskal-Katona theorem which gives a lower bound on $|\delta(\mathcal{A})|$ and describe related notions of colex order and compression operators on set families.

### February 1, 2019

### 9:00 AM

### AP&M 5402

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