challenges / min-triangles-fixed-edges
Fewest triangles for a given number of edges
The Erdős–Rademacher problem at finite n: a graph on n vertices with exactly e edges and as few triangles as possible, for nine (n, e) instances with n up to 50 above the Mantel threshold. Scored against the Lovász–Simonovits construction, proven optimal for large n (Liu–Pikhurko–Staden 2020) and conjectured for all n; the asymptotic form is AlphaEvolve problem 46 (Razborov's theorem).
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# Fewest triangles for a given number of edges
## Goal
`build.py` exposes `build(n: int, e: int, time_budget: float, seed: int) -> list[tuple[int, int]]`:
a simple graph on the vertex set `range(n)` with exactly `e` distinct edges (no loops, no repeated
edges, either endpoint order). Minimise the number of triangles.
This is the Erdős–Rademacher problem. Mantel says `e <= n^2/4` allows zero triangles; every
instance here has more edges than that, so triangles are forced, and the question is how few.
Asymptotically the answer is Razborov's theorem (2008): the minimum triangle density at edge
density `rho` is attained by complete multipartite graphs with all parts equal but one smaller
part. That asymptotic curve is AlphaEvolve problem 46 ("minimal triangle density in graphs",
Section 6.27 of arXiv:2511.02864), where AlphaEvolve matched the known optimum. The exact finite
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