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1017 of 1221 statements are formalized in Lean (Formal Conjectures) · 107 carry an Erdős prize · 3 are independent of ZFC and 9 more can't be proved, or can't be disproved, from ZFC alone · “open” counts the computable and Lean-checked ones too.
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Beyond ZFC · no Lean solution · 12 problems✕ clear
For every \(x \in \mathbb{R}\) let \(A_x \subset \mathbb{R}\) be a bounded set with outer measure \(< 1\). Must there exist an infinite independent set, that is, some infinite \(X \subseteq \mathbb{R}\) such that \(x \notin A_y\) for all…
Let \(\mathfrak{m}\) be an infinite cardinal and \(G\) be a graph with chromatic number \(\mathfrak{m}\). Is it true that, for every infinite cardinal \(\mathfrak{n}< \mathfrak{m}\), there exists a subgraph of \(G\) with chromatic number…
Let \(\mathfrak{m}\) be an infinite cardinal with \(\aleph_0 < \mathfrak{m} < \mathfrak{c} = 2^{\aleph_0}\). Let \(\{f_\alpha\}\) be a family of entire functions such that, for every \(z_0 \in \mathbb{C}\), there are at most…
Is it true that \(\omega_1^2 \not\to (\omega_1^2, 3)^2\)? A problem of Erdős and Hajnal. Hajnal [Ha71] proved this assuming the continuum hypothesis. The source's displayed relation has blue target \(3\); its introductory \(k\) does not…
**Erdős Problem 1171.** Is it true that for all finite \(k < \omega\), \[\omega_1^2 \to (\omega_1 \cdot \omega, \underbrace{3, \ldots, 3}_{k})^2_{k+1}?\] In any \((k+1)\)-coloring of the pairs of the complete graph on \(\omega_1^2\),…
**Erdős Problem 1174, part (i)** [Va99, 7.91]. A problem of Erdős and Hajnal: does there exist a graph \(G\) with no \(K_4\) such that every edge colouring of \(G\) with countably many colours contains a monochromatic \(K_3\)? For…
Let \(G\) be a graph with chromatic number \(\aleph_1\). Is it true that there is a colouring of the edges with \(\aleph_1\) many colours such that, in any countable colouring of the vertices, there exists a vertex colour containing all…
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