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Claim. Every graph with $n$ vertices and more than $n^2/4$ edges has an edge lying in at least $n/6$ triangles, the statement of Problem 905. The site credits the statement to Edwards as an independent proof, marked unpublished. The proof the sources credit is an unpublished manuscript. Erdős's 1993 survey (card, Chapter V, problem 4, p. 344) writes that Edwards proved the conjecture but that his proof was never published, its reference [47] describing a manuscript obtainable from a colleague at Memphis State University; Fox and Loh, in a refereed paper (Combinatorica 32 (2012); p. 2 of the preprint, recorded on the card for Problem 80), attest the same result as their reference [4], a 1977 unpublished manuscript, which gives this page its year; no source read gives a month or day. Bollobás and Nikiforov's refereed paper Books in graphs (European J. Combin. 26 (2005); [BoNi05] on the problem page, cited from arXiv:math/0405080v1) says that the conjecture was proved by Edwards in an unpublished manuscript, its reference [3], C. S. Edwards, A lower bound for the largest number of triangles with a common edge, 1977, and attributes to him its Corollary 2, $\mathrm{bk}(G)\ge2m/n-n/3$ for every graph with $n$ vertices and $m>n^2/4$ edges, whose proof gives the statement with a surplus. A separate posting is C. S. Edwards, The largest number of triangles with a common edge in a graph, Colloques internationaux C.N.R.S. 260 (Problèmes combinatoires et théorie des graphes, Orsay 1976), Paris (1978), 123--126, as reference [3] of Khadzhiivanov's 1988 paper gives it (card, pp. 47--48): five theorems announced without proofs, which the 1988 paper says did not appear in the following ten years, and which it reads as not solving the conjecture: all of them concern Erdős's conjecture but, in the paper's words in translation, bring no shift toward its solution (p. 48), and the theorem Edwards calls his main one has the negation of the conjecture among its hypotheses. Whether the manuscript and the announcement carry the same argument is not known; neither text is held, and no text of the proof is held or located.

Depends on. Nothing in this wiki.

Acceptance. The reviewed evidence is the documented acceptance of the site's curator (T. F. Bloom), independent of the claimant, whose label PROVED (LEAN) settles the problem and whose commentary credits this proof, marked unpublished, beside the 1979 note; Erdős's 1993 report, Fox and Loh's refereed attestation and Bollobás and Nikiforov's refereed attribution of their Corollary 2 document that the result was announced and believed by independent experts. No refereed or formalized evidence exists for it: the proof was never published, no copy was found, and the one source that examined Edwards's 1978 announcement, Khadzhiivanov's 1988 paper, finds no solution in it; that critique concerns the announcement, not the 1977 manuscript, which the three attestations credit. The problem's standing is derived from this page together with the accepted claim pages Khadzhiivanov and Nikiforov and Bollobás and Nikiforov.