Caccetta–Häggkvist for directed triangles

• Slug: caccetta-haggkvist-k3
• List: P36
• Solver: SuperGrok CLI grok-4.6 --reasoning-effort xhigh
• Status: residue — leftover exact cubes through n=152 stored and replayed (q42 n=152: 49 cubes, verify_range 0 failures; summaries in-tree, DRATs local). F₄ certificate c=0.34640 (CKLS 2015 fork; HKN 0.3465; prior stored 0.34645). First leftover hole n=153. Did not beat 0.3388. Conjecture 1/3 open.
• Area: Directed graph theory
• Sources: AIM Caccetta–Häggkvist summary; Egres open problem page
• Started: 2026-08-17

Statement

The first open case asks whether every n-vertex digraph of minimum out-degree at least n/3 contains a directed triangle.

Tonight

A certified small-order obstruction, an improved numerical out-degree threshold with an independently checkable certificate, or a documented incomplete search. Isolated random-graph statistics are not a new bound. Fetch the current published threshold before searching.