M. Balko

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math.CO2025

Characterisation of the big Ramsey degrees of the generic partial order

Martin Balko, David ChodounskÃ½, Natasha Dobrinen +4

As a result of 33 intercontinental Zoom calls, we characterise big Ramsey degrees of the generic partial order. This is an infinitary extension of the well known fact that finite p…

math.CO2025

On forbidden configurations in point-line incidence graphs

Martin Balko, NÃ³ra Frankl

The celebrated SzemerÃ©di--Trotter theorem states that the maximum number of incidences between $n$ points and $n$ lines in the plane is $O (n^{4/3})$ , which is asymptotically tight…

math.CO2025

Crossing and non-crossing families

Todor AntiÄ, Martin Balko, Birgit Vogtenhuber

For a finite set $P$ of points in the plane in general position, a \emph{crossing family} of size $k$ in $P$ is a collection of $k$ line segments with endpoints in $P$ that are pai…

math.CO2025

Faces in rectilinear drawings of complete graphs

Martin Balko, Anna BrÃ¶tzner, Fabian Klute +1

We initiate the study of extremal problems about faces in convex rectilinear drawings of~ $K_{n}$ , that is, drawings where vertices are represented by points in the plane in convex po…

math.CO2025

On the Uncrossed Number of Graphs

Martin Balko, Petr HlinÄnÃ½, TomÃ¡Å¡ MasaÅÃk +3

Visualizing a graph $G$ in the plane nicely, for example, without crossings, is unfortunately not always possible. To address this problem, MasaÅÃk and HlinÄnÃ½ [GD 2023] recent…

math.CO2025

A Survey on Ordered Ramsey Numbers

Martin Balko

The ordered Ramsey number of a graph $G^{<}$ with a linearly ordered vertex set is the smallest positive integer $N$ such that any two-coloring of the edges of the ordered complete g…