# Infinite Graphs

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- Menger's theorem for infinite graphs
- Ordinal Ramsey: For which \(\alpha\) does \(\omega^{\alpha} → (\omega^{\alpha}, 3)^{2}\)? (Hajnal) ($1000)
- Ordinal Ramsey: If \(\alpha → (\alpha, 3)^{2},\) then \(\alpha → (\alpha, 4)^{2}\) (Hajnal)
- Ordinal Ramsey: \(\omega_{1} → (\alpha, 4)^{3}\) for \(\alpha < \omega_{1}\)
- Ordinal Ramsey: \(\omega_{1}^{2} → (\omega_{1}^{2}, 3)^{2}\)
- Ordinal Ramsey: \(\omega_{3} → (\omega_{2} + 2)^{3}_{\omega}\)
- Almost-bipartite graphs with infinite chromatic number (Hajnal, Szemerédi) ($250)
- Another problem on almost-bipartite graphs with infinite chromatic number
- Uncountable-chromatic graphs have common \(4\)-chromatic subgraphs (Hajnal)
- Infinite \(K_{4}\)-free graphs are the union of triangle-free countable graphs (Hajnal) ($250)
- The harmonic sums of odd-cycle lengths diverges for graphs of infinite chromatic number (Hajnal)
- There is an uncountable-chromatic graph \(G\) so that the size of the smallest \(n\)-chromatic subgraph grows arbitrarily slowly (Hajnal, Szemerédi)
- Infinite graphs have infinite paths or arbitrary independent sets (Hajnal, Milner)
- Down-up matchings in infinite graphs (Larson)