challenges / turan-tetrahedron
Tetrahedron-free 3-graphs, maximum size
Turán's 1941 problem at finite n: the largest 3-uniform hypergraph on n vertices with no tetrahedron K_4^(3), for n in {9, 11, 13, 16, 20, 24}. Scored against Turán's construction, which is proven optimal for n <= 13 and conjectured optimal beyond (AlphaEvolve problem 37).
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# Tetrahedron-free 3-graphs, maximum size
## Goal
`build.py` exposes `build(n: int, time_budget: float, seed: int) -> list[tuple[int, int, int]]`:
a 3-uniform hypergraph on the vertex set `range(n)`, given as a list of triples of distinct vertices
(any order inside a triple, no repeated triples). No four vertices may carry all four of their
triples, i.e. the hypergraph contains no tetrahedron `K_4^(3)`. Maximise the number of triples.
This is Turán's 1941 problem, the oldest open question in extremal hypergraph theory. Turán's
construction splits the vertices into three parts `V0, V1, V2` as equal as possible and takes every
triple with one vertex in each part, plus every triple with two vertices in `Vi` and one in
`V(i+1 mod 3)`. Asymptotically that is `5/9` of all triples; Razborov's flag-algebra upper bound is
`0.5616...`. Nobody has ever found a tetrahedron-free 3-graph with more triples than Turán's
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