Smooth graphs

Random graphs in the monadic theory of order

We continue the works of Gurevich-Shelah and Lifsches-Shelah by showing that it is consistent with ZFC that the first-order theory of random graphs is not interpretable in the monadic theory of all chains. It is provable from ZFC that the theory of random graphs is not interpretable in the monadic second order theory of short chains (hence, in the monadic theory of the real line).

ZFC independence and subset sum

We study recursively defined functions associated with directed graphs on the k dimensional nonnegative integral lattice. The existence of certain combinatorial structures associated with these function classes are shown to be independent of the ZFC axioms of mathematics. These structures, in a natural way, give rise to sets of instances to the subset sum problem. We use this connection to make some observations about ZFC independence and the subset sum problem.

No universal graphs at uncountable regular cardinals

Assuming the existence of a strong cardinal, we find a model of ZFC in which for each uncountable regular cardinal $λ,$ there is no universal graph of size $λ$.

$K_{\omega_1}$和$K_{\omega,\omega_1}$上的Maker-Breaker博弈

Maker-Breaker games on $ K_{ω_1}$ and $K_{ω,ω_1}$

该研究探讨两类无穷图上的Maker-Breaker博弈，证明博弈结果依赖于集合论公理框架，给出不同公理体系下双方的必胜策略结论。

Orientations of graphs with uncountable chromatic number

Motivated by an old conjecture of P. Erdős and V. Neumann-Lara, our aim is to investigate digraphs with uncountable dichromatic number and orientations of undirected graphs with uncountable chromatic number. A graph has uncountable chromatic number if its vertices cannot be covered by countably many independent sets, and a digraph has uncountable dichromatic number if its vertices cannot be covered by countably many acyclic sets. We prove that consistently there are digraphs with uncountable dichromatic number and arbitrarily large digirth; this is in surprising contrast with the undirected ca…

On the number of non-isomorphic subgraphs

Let K be the family of graphs on omega_1 without cliques or independent subsets of size omega_1 . We prove that: 1) it is consistent with CH that every G in K has 2^{omega_1} many pairwise non-isomorphic subgraphs, 2) the following proposition holds in L: (*) there is a G in K such that for each partition (A,B) of omega_1 either G cong G[A] or G cong G[B], 3) the failure of (*) is consistent with ZFC.