The board / set theory / #598
Erdős Problem #598
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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Let \(\kappa = (2^{\aleph_0})^+\). This is the successor cardinal of the continuum.
**Erdős Problem 598:** Let \(m\) be an infinite cardinal and \(\kappa\) be the successor cardinal of \(2^{\aleph_0}\). Can one colour the countable subsets of \(m\) using \(\kappa\) many colours so that every \(X \subseteq m\) with \(|X| = \kappa\) contains subsets of all possible colours?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open. The authoritative wording is on erdosproblems.com/598.Prizenone
Topicsset theory, ramsey theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/598
Attempts
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Relative independence is verified modulo Garti--Hayut and Jech; additionally, any first bad cardinal is proved regular and \(\aleph_0\)-closed, sharpening the linked threshold note, while the exact co
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