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Claim. $k(3,4)=21$ for Problem 112: every directed graph on $21$ vertices has three independent vertices or a transitive tournament on four vertices, and some directed graph on $20$ vertices has neither. The lower bound is the circulant on $\mathbb Z_{20}$ with connection set $\{2,4,5,7,11,14\}$, arcs $i\to j$ when $j-i\bmod 20$ lies in the set, which the posting says has no independent 3-set and no transitive tournament on four vertices, checked by a short script that shares no code with the search. The upper bound is not one exhaustive SAT run: the direct $21$-vertex instance did not terminate. It is a case split. For a vertex $v$, the out- and in-neighborhoods each have independence number at most $2$ and no transitive triple, so each has at most $8$ vertices and one of a listed set of isomorphism types, and the non-neighbors of $v$ form a tournament with no transitive 4-set, so there are at most $7$ of them. A counting lemma on a vertex of maximum out-degree $a$ forces $a\ge7$ and leaves $367$ of $640$ cases. Each case was refuted by the SAT solver kissat with a DRAT proof checked by drat-trim; the per-case proofs were deleted after checking, and the repository keeps their SHA-256 digests and the commands that regenerate them, with four small proofs kept. In the site's letters this is $r(I_3,L_4)$ of Ihringer, Rajendraprasad and Weinert, whose recursion gives $k(3,4)\le25$.

Covers. The single value $k(3,4)=21$. The same posting's brackets $34\le k(4,4)\le50$ and $31\le k(3,5)\le55$ and its recursion $k(n,m)\le2k(n,m-1)+k(n-1,m)-1$ are bounds and settle no instance; the values $k(3,3)=9$ and $k(4,3)=15$ that the method recovers are reproductions of published results.

Depends on. Nothing in this wiki.

Standing. Claimed. The result was posted as a comment in the site's discussion thread on 22 September 2026 by Muhamadiev Faridun, with the public repository created the same day and linked above at its head commit. The comment and the repository's README state that the work was done with AI assistance (Claude Opus 5) under the author's direction. The README says that the repository is a computation and not a solution, that the problem stays open, and that the upper bound is not machine-checked as a whole: the counting lemma is a hand proof and load-bearing (the other $273$ cases were never run), the soundness of the per-case symmetry breaking is a hand argument that drat-trim does not check, and the enumeration of the neighborhood blocks is the author's own code, not cross-checked against an independent tool. No review or rerun by anyone other than the author is known.