Trees, ladders and graphs

We introduce a new method to construct uncountably chromatic graphs from non special trees and ladder systems. Answering a question of P. Erd\H{o}s and A. Hajnal from 1985, we construct graphs of chromatic number $\omega_1$ without uncountable $\omega$-connected subgraphs. Second, we build triangle free graphs of chromatic number $\omega_1$ without subgraphs isomorphic to $H_{\omega,\omega+2}$.Comment: 23 pages, 2 figures, submitted to the Journal of Comb. Theory Series

On the growth rate of chromatic numbers of finite subgraphs

We prove that, for every function $f:\mathbb{N} \rightarrow \mathbb{N}$, there is a graph $G$ with uncountable chromatic number such that, for every $k \in \mathbb{N}$ with $k \geq 3$, every subgraph of $G$ with fewer than $f(k)$ vertices has chromatic number less than $k$. This answers a question of Erd\H{o}s, Hajnal, and Szemeredi.Comment: 10 page

Uncountable dichromatic number without short directed cycles

A. Hajnal and P. Erd\H{o}s proved that a graph with uncountable chromatic number cannot avoid short cycles, it must contain for example $C_4$ (among other obligatory subgraphs). It was shown recently by D. T. Soukup that, in contrast of the undirected case, it is consistent that for any $n<\omega$ there exists an uncountably dichromatic digraph without directed cycles shorter than $n$. He asked if it is provable already in ZFC. We answer his question positively by constructing for every infinite cardinal $\kappa$ and $n<\omega$ a digraph of size $2^{\kappa}$ with dichromatic number at least $\kappa^{+}$ which does not contain directed cycles of length less than $n$ as a subdigraph.Comment: 3 pages, 1 figur

Analytic digraphs of uncountable Borel chromatic number under injective definable homomorphism

We study the analytic digraphs of uncountable Borel chromatic number on Polish spaces, and compare them with the notion of injective Borel homomorphism. We provide some minimal digraphs incomparable with G 0. We also prove the existence of antichains of size continuum, and that there is no finite basis. 2010 Mathematics Subject Classification. 03E15, 54H0

Cycle reversions and dichromatic number in tournaments

We show that if $D$ is a tournament of arbitrary size then $D$ has finite strong components after reversing a locally finite sequence of cycles. In turn, we prove that any tournament can be covered by two acyclic sets after reversing a locally finite sequence of cycles. This provides a partial solution to a conjecture of S. Thomass\'e.Comment: 23 pages, first public version. Comments are very welcom

Finite subgraphs of uncountably chromatic graphs

It is consistent that for every monotonically increasing function f:omega->omega there is a graph with size and chromatic number aleph_1 in which every n-chromatic subgraph has at least f(n) elements (n >= 3). This solves a $250 problem of Erdos. It is also consistent that there is a graph X with Chr(X)=|X|= aleph_1 such that if Y is a graph all whose finite subgraphs occur in X then Chr(Y)<=aleph_2 (so the Taylor conjecture may fail)

On the growth rate of dichromatic numbers of finite subdigraphs

Chris Lambie-Hanson proved recently that for every function $f:\mathbb{N}\rightarrow \mathbb{N}$ there is an $\aleph_1$-chromatic graph $G$ of size $2^{\aleph_1}$ such that every $(n+3)$-chromatic subgraph of $G$ has at least $f(n)$ vertices. Previously, this fact was just known to be consistently true due to P. Komj\'ath and S. Shelah. We investigate the analogue of this question for directed graphs. In the first part of the paper we give a simple method to construct for an arbitrary $f:\mathbb{N}\rightarrow \mathbb{N}$ an uncountably dichromatic digraph $D$ of size $2^{\aleph_0}$ such that every $(n+2)$-dichromatic subgraph of $D$ has at least $f(n)$ vertices. In the second part we show that it is consistent with arbitrary large continuum that in the previous theorem "uncountably dichromatic" and "of size $2^{\aleph_0}$" can be replaced by "$\kappa$-dichromatic" and "of size $\kappa$" respectively where $\kappa$ is universally quantified with bounds $\aleph_0 \leq \kappa \leq 2^{\aleph_0}$.Comment: 6 page

Extremal triangle-free and odd-cycle-free colourings of uncountable graphs

The optimality of the Erd\H{o}s-Rado theorem for pairs is witnessed by the colouring $\Delta_\kappa : [2^\kappa]^2 \rightarrow \kappa$ recording the least point of disagreement between two functions. This colouring has no monochromatic triangles or, more generally, odd cycles. We investigate a number of questions investigating the extent to which $\Delta_\kappa$ is an \emph{extremal} such triangle-free or odd-cycle-free colouring. We begin by introducing the notion of $\Delta$-regressive and almost $\Delta$-regressive colourings and studying the structures that must appear as monochromatic subgraphs for such colourings. We also consider the question as to whether $\Delta_\kappa$ has the minimal cardinality of any \emph{maximal} triangle-free or odd-cycle-free colouring into $\kappa$. We resolve the question positively for odd-cycle-free colourings.Comment: 16 page

A determinacy approach to Borel combinatorics

We introduce a new method, involving infinite games and Borel determinacy, which we use to answer several well-known questions in Borel combinatorics.Comment: Minor corrections and some reorganization of section

Counterexamples to Hedetniemi's conjecture

The chromatic number of $G\times H$ can be smaller than the minimum of the chromatic numbers of finite simple graphs $G$ and $H$.Comment: 3 pages, minor corrections, a version accepted for publicatio