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When is it true that if $G$ is isomorphic to a spanning subgraph of $H$ and $H$ is isomorphic to a spanning subgraph of $G$, then $G$ is isomorphic to $H$? Clearly this is true if $G$ and $H$ are fini …

After the discussion above, here is what I think is the cleanest proof and it has the property that $f$ is bijection (unless there is an edge of order 1). If there is an edge of order 1, then we must …

How about a graph $G=(V,E)$ consisting of infinitely many isolated vertices and infinitely many disjoint edges. Like the random graph it has the property that for all $e\in E$, $G\simeq (V, E\setminu …

My first thought for the case where $|V|\leq \aleph_0$ is that surely the Rado graph can be constructed as an induced subgraph of $([\omega]^{\omega}, E)$ (since the Rado graph contains a copy of ever …

This question was motivated by a discussion here and is related to a previous question here. Let $\kappa$ and $\lambda$ be cardinals such that $0<\lambda\leq \kappa$. Let $G=(A\cup B, E)$ be a bipart …

(Just making my comment an answer as suggested.) If every finite subgraph of $G$ satisfies Kőnig's Property, then $G$ has no odd cycles and is thus bipartite. Aharoni (König's Duality Theorem For Infi …