On universal graphs with forbidden topological subgraphs (Q1068096)

A graph \(G^*\) is called universal in a class \(\bar G\) of countable graphs if it contains a copy of every G in \(\bar G.\) The graph \(G^*\) is called strongly universal if every G in \(\bar G\) is isomorphic to an induced subgraph of \(G^*\). For each pair n and m of positive integers, let \(\bar G(\)n,m) represent the class of all countable graphs not admitting \(K^{n,m}\) as a subdivision. Halin has asserted that for n,m sufficiently large \(\bar G(\)n,m) has no universal element. The author proves this conjecture. He also shows that \(\bar G(\)2,3) contains a universal element but no strongly universal one.