The board / graph theory / #571
Erdős Problem #571
PROVED LEANdatabase updated 2026-09-03
The statement as posed is true, and a proof exists. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Show that for any rational \(\alpha \in [1,2)\) there exists a bipartite graph \(G\) such that \[\mathrm{ex}(n;G)\asymp n^{\alpha}.\] The proof constructs balanced rooted models for all rational parameters. Its upper-bound closure replaces old edges by paths of arbitrary length, adds two color-class hubs, and commutes with positive rooted powers.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/571.Prizenone
Topicsgraph theory, turan number
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/571
Attempts
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combining the 2026 JLY theorem with KKL densification proves the family (5), including \(29/18\), and certifies all 46 reduced rationals of denominator at most 12; the first gap in the audited suite i
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