Michał Lasoń

Triangle-free intersection graphs of line segments with large chromatic number

In the 1970s, Erdos asked whether the chromatic number of intersection graphs of line segments in the plane is bounded by a function of their clique number. We show the answer is no. Specifically, for each positive integer

On the toric ideal of a matroid

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Triangle-Free Geometric Intersection Graphs with Large Chromatic Number

, we construct a triangle-free family of line segments in the plane with chromatic number greater than

On the full, strongly exceptional collections on toric varieties with Picard number three

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Coloring hypergraphs induced by dynamic point sets and bottomless rectangles

. Our construction disproves a conjecture of Scott that graphs excluding induced subdivisions of any fixed graph have chromatic number bounded by a function of their clique number.

Coloring Intersection Graphs of Arc-Connected Sets in the Plane

Abstract Describing minimal generating set of a toric ideal is a well-studied and difficult problem. In 1980 White conjectured that the toric ideal associated to a matroid is equal to the ideal generated by quadratic binomials corresponding to symmetric exchanges. We prove Whites conjecture up to saturation, that is that the saturations of both ideals are equal. In the language of algebraic geometry this means that both ideals define the same projective scheme. Additionally we prove the full conjecture for strongly base orderable matroids.

Indicated coloring of matroids

Several classical constructions illustrate the fact that the chromatic number of a graph may be arbitrarily large compared to its clique number. However, until very recently no such construction was known for intersection graphs of geometric objects in the plane. We provide a general construction that for any arc-connected compact set

The coloring game on matroids

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