Smallest tournament with capacity above its transitive number
Follow the latest progress by email.
For a tournament T, compare its directed capacity C(T) with the size t(T) of its largest transitive subtournament. What is the smallest vertex count at which C(T)>t(T)?
Why it matters and what counts as progress
Why it matters: The minimum-order problem sharpens a known failure of equality by locating the first possible finite witness.
What counts as progress: A smaller tournament with a rigorous capacity lower bound above t(T), and a classification excluding all smaller orders; exact capacity need not be known.
How the problem developed
- 1998 · Capacity result [1]
Noga Alon Developed the capacity of digraphs and constructed tournaments whose capacity grows faster than the size of their largest transitive subtournament, establishing that a gap can occur.
- October 2026 · Minimum order claim [2]
JD-Jones-ASES contributors A Lean-backed project reports that the Paley tournament T7 has capacity above its transitive number and that all tournaments of order at most six have equality, making seven the least order with a gap.
AI contribution
Claude-assisted search finds the smallest tournament with capacity above its transitive number
The specified seven-vertex induced Paley tournament has transitive number 4 and capacity between √18 and 5; every tournament on at most six vertices has capacity equal to its transitive number, so seven is minimal.
Sources
Website timeline
Publication dates and editorial changes.
Claude-assisted search finds the smallest tournament with capacity above its transitive number