---

Claim. Every graph on $n$ vertices with more than $n^2/4$ edges has an edge lying on at least $n/6$ triangles. For the function $f_c(n)$ of Problem 80 this gives $f_c(n)\ge n/6$ for every fixed $c>1/4$, without the hypothesis that every edge lies in a triangle. So $f_c(n)>n^\epsilon$ holds for every $\epsilon<1$ and all large $n$ in that range, $f_c(n)\gg\log n$ holds there, and $f_c(n)=\Theta(n)$ there, since a book has at most $n-2$ pages. This is the theorem of Problem 905, which the site credits to Edwards (unpublished) and, independently, to the 1979 note of Khadzhiivanov and Nikiforov, its source key [KhNi79]. The note is not held; the proof relied on is Khadzhiivanov's 1988 account, khadzhiivanov_1988_maximal_number_triangles_common_edge, whose Corollary 3 (p. 45) proves it with strict inequality, from the paper's Theorem 1, $(3t+\bar t)\hat t\ge nt$, and its Lemma 4; the account's own Corollary 4 extends it to graphs with at least $[n^2/4]$ edges and a triangle, which the problem page uses for $c=1/4$; Corollary 5 there gives the exact minimum $\lceil n/6\rceil$ of the largest book over the $n$-vertex graphs with at least $[n^2/4]$ edges and a triangle, so the constant $1/6$ cannot be raised, and Theorem 2 (p. 43) gives a book of size at least $\frac{r-2}rn$ at every density at least $\frac{r-1}{2r}$. Fox and Loh and Potechin cite the bound (each on p. 2 of the preprint), Fox and Loh as the reason the range $c<1/4$ of their theorem is best possible, and Erdős's 1988 passage, quoted on the problem page, calls the linear bound for $c>1/4$ well known.

Covers. The range $c>1/4$ of Problem 80: there $f_c(n)\ge n/6$, so the properties both closing questions ask for, $f_c(n)>n^\epsilon$ for some $\epsilon>0$ and $f_c(n)\gg\log n$, hold, and $f_c(n)=\Theta(n)$, since $n/6\le f_c(n)\le n-2$, the upper bound being trivial. Outside this page: the case $c=1/4$, given by Corollary 4 of Khadzhiivanov's 1988 account and recorded on the problem page; and the range $c<1/4$, where the first closing question fails, so that Fox and Loh 2012 answers it no, and where the page-level estimate and the logarithmic question are open.

Depends on. Corollary 3 of Khadzhiivanov's 1988 account, the proof relied on.

Acceptance. The site's curator, T. F. Bloom, labels Problem 905, whose statement is the claim above, PROVED (LEAN) and credits it to Edwards and, independently, to Khadzhiivanov and Nikiforov, and records the bound $f_c(n)\ge n/6$ for $c>1/4$ in this problem's commentary with a pointer to Problem 905 (both pages last edited 7 April 2026, accessed 2026-09-18). That label settles Problem 905, not Problem 80 or a part of it, and the commentary on Problem 80, which the site labels OPEN, is not acceptance, so no reviewed evidence is listed. The Lean proof behind the site's suffix is the file in Boris Alexeev's repository linked above at the commit of 15 September 2026, which declares itself a formalization of the 1979 note's result, the claim above; it is linked, not built in this corpus, so it is no formalized evidence here. Refereeing is not documented: the 1979 note appeared in C. R. Acad. Bulgare Sci. 32 (1979), 1315--1318, and the 1988 account in Annuaire Univ. Sofia, Fac. Math. Inform. 82 (1988), 37--49 (the journal's article record, linked above, lists volume 82, number 1, pp. 37--49), but no record shows that either venue refereed them, so refereed is not listed. With no evidence listed, the claim stands as claimed. The 1988 text's Corollary 3 is covered at claims-checked depth, and the proofs of its Theorem 1 and Lemma 4 at structure depth, as the library card records; nothing is independently reviewed in this corpus. Edwards's announcement (Colloques internationaux C.N.R.S. 260, 1978) is not held, and the 1988 account reports that the proofs it announced were never published, so Edwards's independent proof is disclosed here and not relied on.

Dating. The page is dated by the year of the 1979 note, the source the site cites, which the 1988 account's reference list gives as Dokl. BAN 32 (1979), no. 10, 1315--1318; the month and day are placeholders. The proof relied on is the 1988 account.