---

Claim. D. Neiman, J. Mackey and M. J. H. Heule, Tighter bounds on directed Ramsey number $R(7)$, Graphs Combin. 38 (2022), no. 5, Paper No. 156, prove $34\le R(7)\le47$ (Section 5 of the library's source card). The upper bound says that every tournament on $47$ vertices contains a transitive subtournament on $7$ vertices, so $f(n)\ge7$ for every $n\ge47$, while the formula of Problem 1216 gives $\lfloor\log_2n\rfloor+1=6$ for $32\le n\le63$; the formula fails for $47\le n\le63$, values that include some, $47\le n\le53$, below the reach of the Sánchez-Flores bounds. The lower bound is an explicit $33$-vertex tournament with no transitive $7$-subtournament, so $f(33)=6$. Both bounds are computer-assisted: the upper bound is a SAT-based case analysis of the in- and out-degrees of a $47$-vertex tournament with no transitive $7$-subtournament, built on the authors' classification of the tournaments on $23$ to $25$ vertices with no transitive $6$-subtournament.

Depends on. Reid and Parker 1970 for $R(6)\le28$ (their Corollary 1 at $k=6$), which bounds every in-degree of a tournament with no transitive $7$-subtournament by $27$; the paper cites $R(6)=28$ to Sánchez-Flores 1994.

Source. The page is dated by the first arXiv posting, 2 November 2020; the journal version was published online on 9 September 2022 (per Crossref). The locators of the source card are to the NSF author manuscript.

Acceptance. Refereed: Graphs and Combinatorics 38 (2022), no. 5, Paper No. 156. The site's commentary does not cite this paper, so no reviewed evidence is listed. The computations are not replayed in this corpus.