Answer by Fedor Petrov for Universal graph
I think that the answer is negative.Assume that such graph $G$ on the vertex set $\{v_1,v_2,\ldots\}$ exists. We construct our not-embeddable graph $H$ on $\{1,2,\ldots\}$ by steps. On the $i$-th step...
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A connected (and infinite) graph $U$ will be called $n$-universal if any connected graph with degree $\leqslant n$ admits an embedding in $U$.Is there a 3-universal graph with bounded degree?
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