A tree--arrowing graph

Collapsible Pushdown Graphs of Level 2 are Tree-Automatic

We show that graphs generated by collapsible pushdown systems of level 2 are tree-automatic. Even when we allow $ε$-contractions and add a reachability predicate (with regular constraints) for pairs of configurations, the structures remain tree-automatic. Hence, their FO theories are decidable, even when expanded by a reachability predicate. As a corollary, we obtain the tree-automaticity of the second level of the Caucal-hierarchy.

The Uncountable Hadwiger Conjecture and Characterizations of Trees Using Graphs

We prove that the existence of a non-special tree of size $λ$ is equivalent to the existence of an uncountably chromatic graph with no $K_{ω_1}$ minor of size $λ$, establishing a connection between the special tree number and the uncountable Hadwiger conjecture. Also characterizations of Aronszajn, Kurepa and Suslin trees using graphs are deduced. A new generalized notion of connectedness for graphs is introduced using which we are able to characterize weakly compact cardinals.

树分解与可达性在Borel图可树化中的应用

Applications of tree decompositions and accessibility to treeability of Borel graphs

本文提出Borel方式处理Borel图分支树分解的框架，引出可达性概念与可树化准则，证明相关Borel等价关系的测度可树化等结果，推广前人结论并给出性质(T)等价关系的图化限制。