##### Department of Mathematics,

University of California San Diego

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### PhD Defense

## Ji Zeng

## Variation of no-three-in-line problem

##### Abstract:

The famous no-three-in-line problem by Dudeney more than a century ago asks whether one can select 2n points from the grid $[n]^2$ such that no three are collinear. We present two results related to this problem. First, we give a non-trivial upper bound for the maximum size of a set in $[n]^4$ such that no four are coplanar. Second, we characterize the behavior of the maximum size of a subset such that no three are collinear in a random set of $\mathbb{F}_q^2$, that is, the plane over the finite field of order q. We discuss their proofs and related open problems.

### May 20, 2024

### 10:45 AM

APM 7218

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