On April 14, 2021, István Tomon from the ETH Zurich gave an online talk at the Virtual Discrete Math Colloquium on the size of the maximum clique, the size of the maximum independent set, and the chromatic number of a semilinear graph of bounded complexity. The title of his talk was “Ramsey properties of semilinear graphs“.

## István Tomon, Ramsey properties of semilinear graphs

A graph $G$ is semilinear of bounded complexity if the vertices of $G$ are elements of $\mathbb{R}^{d}$, and the edges of $G$ are defined by the sign patterns of $t$ linear functions, where $d$ and $t$ are constants. In this talk, I will present several results about the symmetric and asymmetric Ramsey properties of semilinear graphs. Some interesting instances of such graphs are intersection graphs of boxes, interval overlap graphs, and shift graphs, so our results extend several well known theorems about the Ramsey and coloring properties of these geometrically defined graphs.