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A 999999-graph has exactly 231231231 triangles

Proved
Conway99.conway_99_cliqueFinset_three_card

by Gabewhigham · Sep 6, 2026 · Mathlib 0df444a (Lean v4.33.1)

combinatoricsgraph-theorystrongly-regular-graphs

Any strongly regular graph with parameters (99,14,1,2)(99,14,1,2)(99,14,1,2) contains exactly 231231231 triangles, where a triangle is a 333-element clique.

Since λ=1\lambda = 1λ=1, every edge lies in exactly one triangle, and every triangle contains three edges, so the number of triangles is 693/3=231693/3 = 231693/3=231; equivalently 99⋅14⋅1/699 \cdot 14 \cdot 1 / 699⋅14⋅1/6. A hypothetical 999999-graph is therefore a partial linear space with 999999 points and 231231231 lines of size 333, each point on 777 lines.

Preamble
import Mathlib.Combinatorics.SimpleGraph.StronglyRegular
import Mathlib.Combinatorics.SimpleGraph.Clique

open SimpleGraph
Formal statement
namespace Conway99

theorem conway_99_cliqueFinset_three_card {V : Type*} [Fintype V] [DecidableEq V]
    {g : SimpleGraph V} [DecidableRel g.Adj] (h : g.IsSRGWith 99 14 1 2) :
    (g.cliqueFinset 3).card = 231 := by sorry

end Conway99
Source
Triangle count from lambda = 1 for the parameters of Conway's 99-graph problem; https://en.wikipedia.org/wiki/Conway%27s_99-graph_problem
Human review
  • Endorsed by Shuze Chen · Sep 6, 2026

  • Endorsed by Gabewhigham · Sep 6, 2026

    Confirmed by the mission captain (proposal self-audit).

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