Jakub Kozik

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

Asymptotically almost all \lambda-terms are strongly normalizing

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Nonrepetitive colouring via entropy compression

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

A note on random greedy coloring of uniform hypergraphs

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Triangle-Free Geometric Intersection Graphs with Large Chromatic Number

. 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.

Intuitionistic vs. classical tautologies, quantitative comparison

We present a quantitative analysis of various (syntactic and behavioral) prop- erties of random �-terms. Our main results show that asymptotically, almost all terms are strongly normalizing and that any fixed closed term almost never appears in a random term. Surprisingly, in combinatory logic (the translation of the �-calculus into combi- nators), the result is exactly opposite. We show that almost all terms are not strongly normalizing. This is due to the fact that any fixed combinator almost always appears in a random combinator.

Nonrepetitive Choice Number of Trees

A vertex colouring of a graph is nonrepetitive if there is no path whose first half receives the same sequence of colours as the second half. A graph is nonrepetitively ℓ-choosable if given lists of at least ℓ colours at each vertex, there is a nonrepetitive colouring such that each vertex is coloured from its own list. It is known that, for some constant c, every graph with maximum degree Δis cΔ2-choosable. We prove this result with c=1 (ignoring lower order terms). We then prove that every subdivision of a graph with sufficiently many division vertices per edge is nonrepetitively 5-choosable. The proofs of both these results are based on the Moser-Tardos entropy-compression method, and a recent extension by Grytczuk, Kozik and Micek for the nonrepetitive choosability of paths. Finally, we prove that graphs with pathwidth θ are nonrepetitively O(θ2)-colourable.

In the full propositional logic, 5/8 of classical tautologies are intuitionistically valid☆

The smallest number of edges forming an n-uniform hypergraph which is not r-colorable is denoted by m(n,r). Erdős and Lovasz conjectured that m(n,2)=\theta(n 2^n)

Conditional Densities of Regular Languages

. The best known lower bound m(n,2)=\Omega(sqrt(n/log(n)) 2^n) was obtained by Radhakrishnan and Srinivasan in 2000. We present a simple proof of their result. The proof is based on analysis of random greedy coloring algorithm investigated by Pluhar in 2009. The proof method extends to the case of r-coloring, and we show that for any fixed r we have m(n,r)=\Omega((n/log(n))^(1-1/r) r^n) improving the bound of Kostochka from 2004. We also derive analogous bounds on minimum edge degree of an n-uniform hypergraph that is not r-colorable.

Multipass greedy coloring of simple uniform hypergraphs

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