On the number of non-isomorphic subgraphs

Maximal Chains of Isomorphic Subgraphs of Countable Ultrahomogeneous Graphs

For a countable ultrahomogeneous graph G let P(G) denote the collection of domains of subgraphs of G isomorphic to G. The order types of maximal chains in the set P(G) U ø ordered by the inclusion are characterized as: (I) the order types of compact sets of reals having the minimum non-isolated, if G is the Rado graph or the Henson graph H_n, for some n>2; (II) the order types of compact nowhere dense sets of reals having the minimum non-isolated, if G is the union of μdisjoint complete graphs of size ν, where μν=ω.

图的嵌入性与Weihrauch度

Embeddability of graphs and Weihrauch degrees

本文利用有效Wadge可归约性与Weihrauch可归约性，研究固定可数图G的（导出）子图同构判定与查找任务的计算复杂性，并回答了BeMent等人提出的若干公开问题。

On Induced Subgraphs of Finite Graphs not Containing Large Empty and Complete Subgraphs

In their celebrated paper [Ramsey-Type Theorems, Discrete Appl. Math. 25 (1989) 37-52], Erdős and Hajnal asked the following: is it true, that for any finite graph H there exists a constant c(H) such that for any finite graph G, if G does not contain complete or empty induced subgraphs of size at least |V(G)|^c(H), then H can be isomorphically embedded into G ? The positive answer has become known as the Erdős-Hajnal conjecture. In Theorem 3.20 of the present paper we settle this conjecture in the affirmative. To do so, we are studying here the fine structure of ultraproducts of finite sets, s…

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. Erdős and A. Hajnal from 1985, we construct graphs of chromatic number $ω_1$ without uncountable $ω$-connected subgraphs. Second, we build triangle free graphs of chromatic number $ω_1$ without subgraphs isomorphic to $H_{ω,ω+2}$.

ω₁-树的射线膨胀与度为ℵ₁的端

Ray inflations of $ω_1$-trees and ends of degree $\aleph_1$

该研究针对ω₁-树的射线膨胀，证明了两类图的细分或子图同构性结论，并在◇公理下构造了哈林端度猜想的反例。

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 Erdős, Hajnal, and Szemeredi.