Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 147

claims/: The 2 claim pages of Problem 147, one per claimant's result; the problem's standing derives from them.

Statement. If $H$ is bipartite with minimum degree $r$ then there exists $\epsilon=\epsilon(H)>0$ such that

\mathrm{ex}(n;H) \gg n^{2-\frac{1}{r-1}+\epsilon}.

Status. DISPROVED (LEAN). The site's commentary (page last edited 18 January 2026) credits Janzer's rainbow Turán paper [Ja23] with the disproof for even $r\ge4$ and his later paper [Ja23b] with the case $r=3$ ; both are refereed, and each is recorded as an accepted claim on the blow-up page and the 3-regular page. The site's label is DISPROVED (LEAN); the external Lean proof it refers to is linked, unverified, under Formalization below.

Source. erdosproblems.com/147, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #147, https://www.erdosproblems.com/147.

References.

[ErSi84] Erdős, P. and Simonovits, M., Cube-supersaturated graphs and related problems. Progress in graph theory (Waterloo, Ont., 1982) (1984), 203-218.
[Ja23] Janzer, Oliver, Rainbow Turán number of even cycles, repeated patterns and blow-ups of cycles. Israel J. Math. 253 (2023), 813-840.
[Ja23b] Janzer, Oliver, Disproof of a conjecture of Erdős and Simonovits on the Turán number of graphs with minimum degree 3. Int. Math. Res. Not. IMRN (2023), 8478-8494.

Formalization. Statement in formal-conjectures, which carries a formal_proof attribute naming Erdos147.lean in Boris Alexeev's lean-proofs, a refutation of the universal statement through the single witness $C_{12}[2]$ , the bipartite $4$ -regular graph with upper exponent $139/84<5/3$ at minimum degree $4$ ; its header names Janzer as informal author and the AI systems Codex and GPT-5.6 Sol as formal authors, and it is linked on the blow-up claim page. The corpus has not built or audited it, and no acceptance is claimed from it.

Current assessment

Janzer's published Theorem 1.4 supplies the disproof: choosing $0<\eta<1/6$ gives $r=3$ counterexamples with upper exponents incompatible with the proposed lower bound. The linked result records the exponent comparison and same-paper dependency chain, with external inputs separate. This page records neither a dated broader status search nor independent proof-review coverage. The site's label DISPROVED (LEAN) refers to an external, unverified Lean proof that refutes the statement through the minimum-degree- $4$ witness $C_{12}[2]$ , linked above; [Ja23] is credited by the site with the even case $r\ge4$ and has its own claim page, but it remains outside the direct-proof compilation, which rests on [Ja23b] alone. The two claim pages carry the acceptance evidence (refereed publication and the site's credit) from which the frontmatter standing is derived.

Progress

Janzer's [[../library/extremal_graph_theory/janzer_2023_disproof_conjecture_erdos_simonovits_turan_number/theorem_1_4_e147|Theorem 1.4]] constructs, for every $\eta>0$ , a 3-regular bipartite graph $H$ with

\operatorname{ex}(n,H)=O(n^{4/3+\eta}).

For $r=3$ , the proposed lower bound has exponent $3/2+\epsilon(H)$ . Taking $\eta<1/6$ gives an incompatible upper exponent, so this one family directly disproves the universal statement. The linked result page records the exact exponent comparison and the complete same-paper dependency chain, with external inputs stated separately. The published theorem statement independently supplies the status evidence.

Known Results

[[../library/extremal_graph_theory/janzer_2023_disproof_conjecture_erdos_simonovits_turan_number/theorem_1_4_e147|Janzer's direct $r=3$ disproof]] derives the counterexample from the explicit $H_{k,\ell}$ construction and Theorem 1.6.
[[../library/extremal_graph_theory/janzer_2023_disproof_conjecture_erdos_simonovits_turan_number/_index|Source record]] cites arXiv v2 of 8 November 2021, which its locators follow, and the published DOI.

[Ja23]'s Turán bound for the blow-up $C_{2k}[r]$ is a second, independent disproof, recorded on its claim page; the direct-proof compilation covers [Ja23b] only.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (16)

Problem 146 Janzer's blow-up counterexamples for even degree Janzer's 3-regular counterexample problems/extremal_graph_theory/E0147/claims Problem 714 Extremal and Structural Graph Theory Extremal and Structural Graph Theory extremal_graph_theory/erdos_1984_cube_supersaturated_graphs_related_problems Theorem 3 (p. 209): above C_Q n^{8/5} edges there are c E^12/n^16 cubes and c* E^13/n^18 copies of Q*extremal_graph_theory/janzer_2023_disproof_conjecture_erdos_simonovits_turan_number Theorem 1.4 and the direct disproof of Problem 147 extremal_graph_theory/janzer_2023_rainbow_turan_number_even_cycles_repeated Theorem 1.15 (p. 4): ex(n, C_{2k}[r]) = O(n^{2-1/r+1/(k+r-1)}(log n)^{4k/(r(k+r-1))}), disproving the Erdős-Simonovits conjecture for even s >= 4 Display (4.5): T_n(H) < c n^{2−1/(r−1)} for bipartite H whose every induced subgraph has a vertex of degree below r Chapter 10 - Counterexamples to the compactness and degeneracy conjectures for extremal numbers Chapter 10, Theorem 1.2 - Failure of the 2-degenerate bound

Graph