A sharp lower-bound certificate for the first open transitive tournament case

Definitions and mathematical statement

Let $T_k$ be the transitive tournament on $1,\ldots,k$, with an arc $i\to j$ exactly when $i<j$. A subdivision of $T_k$ in a digraph $D$ consists of distinct branch vertices $v_1,\ldots,v_k$ and, for every pair $i<j$, a directed path from $v_i$ to $v_j$. The internal vertices of all these paths are mutually disjoint and none is a branch vertex. Paths may have length one. Extra arcs of $D$ are allowed, since the subdivision is a subgraph and need not be induced.

Mader's conjecture asks whether, for every positive integer $k$, there exists a positive integer $r=r(k)$ such that every finite nonempty simple loopless digraph with outdegree exactly $r$ at each vertex contains a subdivision of $T_k$. There is no restriction on indegrees, and opposite arcs are allowed. Equivalently, one may replace exact outdegree by minimum outdegree at least $r$: from such a digraph one can retain any $r$ outgoing arcs at each vertex and then apply the exact-degree statement.

The requested threshold depends only on $k$, not on the number of vertices or any further structure of the host. Internal vertex disjointness is essential; pairwise edge-disjoint paths would instead define an immersion. Simplicity is also necessary for the intended conjecture, since repeating arcs can create arbitrarily large outdegree on too few vertices to host the required branch vertices.

Short English statement

Does sufficiently large minimum outdegree force a subdivision of every fixed transitive tournament?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20001075 (AIM-COMBINATORICS-0200), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, The Caccetta-Haggkvist conjecture, item 6. Original workshop question underlying AIM-COMBINATORICS-0200. https://aimath.org/WWN/caccetta/caccetta.pdf.