---

Claim. Every graph on $n\ge4$ vertices with at least $2n-3$ edges contains a cycle with a chord, an edge joining two vertices of the cycle that are not consecutive on it. The complete bipartite graph $K_{2,n-2}$ has $2n-4$ edges and no chorded cycle, since each of its cycles is a four-cycle whose chords would join two vertices of one part, so $g_1(n)=2n-4$ for all $n\ge4$ in the notation of Problem 767. This is Pósa's theorem, posed as L. Pósa, Problem 127 (in Hungarian), Mat. Lapok 12 (1961), 254 (the record carries the year only, so this page's name uses its first day). The posting is not held, and the corpus holds no printed proof; the theorem is known through Erdős's reports ([Er64c], p. 36, "For $k=2$ this is Pósa's result" in that paper's indexing; [Er75], pp. 13--14, with the hypothesis $n\ge4$; [Er76b], pp. 191--192, as the least forcing number $g_1(n)=2n-3$), through R. J. Gould, Results and problems on chorded cycles: a survey, Graphs Combin. 38 (2022), article 189, doi:10.1007/s00373-022-02586-9, whose Section 1 states the theorem with the posting as its reference [62] and refers for the solution to L. Lovász, Combinatorial Problems and Exercises (1979), problem 10.2, solution p. 376, and through the preprint of X. Chen and B. Ning (arXiv:2609.15330, reference [19]), which calls it a classical theorem of Pósa and derives the case $k=1$ of its formula from it. Gould reports from the same solution that Czipszer's 1963 answer to Pósa's question gives more: a graph with minimum degree at least $k+2$ has a cycle with $k$ chords at one vertex, the source of the bound $g_k(n)\le(k+1)n$ that the site credits to Czipszer.

Covers. The case $k=1$ of Problem 767: $g_1(n)=(1+1)n-(1+1)^2=2n-4$ for every $n\ge4$, so the conjectured equality holds for $k=1$ from $n=4$ on, with no threshold beyond the four vertices a chorded cycle needs. The cases $k\ge2$ are not covered; they are settled for $n\ge3k+3$ by Jiang on his claim page, and the pending claim of Chen and Ning on their claim page offers an exact threshold.

Acceptance. Reviewed: the site's curator, Thomas Bloom, labels the problem PROVED and, in the problem's commentary, credits Pósa with $g_1(n)=2n-4$ for $n\ge4$; Erdős's papers of 1964 to 1976, Gould's refereed survey and the Chen--Ning preprint credit the theorem to Pósa. Not refereed: the posting is an entry in a problem section, not a refereed paper, and its solution is known through Lovász's exercise book. This corpus supplies no independent proof review. The formal-conjectures statement of the problem, FormalConjectures/ErdosProblems/767.lean (linked at the commit that added it, 2026-09-21), states the theorem as the variant erdos_767.variants.posa, tagged research solved, with a sorry in place of a proof; a statement file is not a formalization and gives no formalized evidence.