##### Department of Mathematics,

University of California San Diego

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### Food for Thought

## Srivatsa Srinivas

## Mama, there goes that set!

##### Abstract:

Suppose that an infinite subset $A$ of the natural numbers is partitioned into finitely many subsets. A property of $A$ that is always inherited by at least one of the elements of the partition is known as a partition regular property. Suppose we have a method of measuring the size of an infinite subset of the natural numbers. A property $P$ is said to be a density property if every infinite subset of the natural numbers which is large enough, according to our yardstick, satisfies $P$. In the qualitative sense, partition regular properties guarantee that order is conserved and density properties guarantee that order is always achieved at a threshold size. We will provide some interesting examples and applications of partition regular properties, density properties, and related ideas.

### November 8, 2019

### 11:00 AM

### AP&M 5402

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