### Subsquares in random Latin squares and rectangles

- Series
- Graph Theory Seminar
- Time
- Tuesday, December 5, 2023 - 15:30 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Alex Divoux – Georgia Tech

A $k \times n$ partial Latin rectangle is \textit{$C$-sparse} if the number of nonempty entries in each row and column is at most $C$ and each symbol is used at most $C$ times. We prove that the probability a uniformly random $k \times n$ Latin rectangle, where $k < (1/2 - \alpha)n$, contains a $\beta n$-sparse partial Latin rectangle with $\ell$ nonempty entries is $(\frac{1 \pm \varepsilon}{n})^\ell$ for sufficiently large $n$ and sufficiently small $\beta$. Using this result, we prove that a uniformly random order-$n$ Latin square asymptotically almost surely has no Latin subsquare of order greater than $c\sqrt{n\log n}$ for an absolute constant $c$. This is joint work with Tom Kelly, Camille Kennedy, and Jasdeep Sidhu.