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Let be minimal such that can be partitioned into classes so that cannot be expressed as a sum of distinct elements from the same class. How fast does grow?
Source: erdosproblems.com/360
An accepted solution exists. Settled in another form, for example when its parts resolve differently or the question is open-ended.
Solved. Alon and Erdős ([AlEr96], refereed) proved
, with ; Vu ([Vu07], refereed) raised the lower bound to
; and Conlon, Fox and Pham ([CFP21], Theorem 1.5, to
appear in J. Eur. Math. Soc.) determined the order of growth, . The site's commentary
records the first two bounds as proved and credits the order of growth to
Conlon, Fox and Pham, on a page labeled SOLVED. The first two are accepted
partial claims
(Alon and
Erdős, Vu), and the
third is the accepted full claim
(claim page (Conlon, Fox and Pham, 2021)), whose acceptance evidence is the site's documented acceptance, the
journal version having no record yet. A 2026 Lean formalization
of the Conlon–Fox–Pham order in Boris Alexeev's public repository, registered by
no outside record, attributes its mathematics to the three authors, so it is a
formalization link on their claim page, neither built nor audited here, and
gives no formalized evidence.