Motivation

A strongly regular graph with parameters $(n,k,\lambda,\mu)$ is a finite simple graph on $n$ vertices in which every vertex has exactly $k$ neighbours, every pair of adjacent vertices has exactly $\lambda$ common neighbours, and every pair of non-adjacent vertices has exactly $\mu$ common neighbours. For most parameter tuples the elementary counting and integrality conditions already decide existence; the interesting cases are those that survive every known feasibility test and still resist construction. The tuple $(99,14,1,2)$ is the smallest such case in the family $\lambda = 1$, $\mu = 2$, and its existence has been open for more than fifty years. John Horton Conway offered $1000 for a resolution, as one of five problems posed at the 2014 DIMACS conference on Challenges of Identifying Integer Sequences (Conway, Five $1,000 Problems (Update 2017)).

Timeline of the problem and of what is known about it:

• 1969/1971 — the parameter set is raised by Norman Biggs in his Southampton lectures (Finite Groups of Automorphisms, LMS Lecture Note Series 6, p. 111).
• 1973 — Berlekamp, van Lint and Seidel construct a strongly regular graph with parameters $(243,22,1,2)$ as the coset graph of the perfect ternary Golay code, settling one of the five feasible parameter tuples in this family.
• 1975 — the existence question appears as Problem 7 (attributed to J. J. Seidel) in R. K. Guy's problem list, The Geometry of Metric and Linear Spaces, Springer LNM 490, pp. 237–238; Conway had worked on it by then.
• 1984 — H. A. Wilbrink, On the $(99,14,1,2)$ strongly regular graph, shows that such a graph cannot be vertex-transitive: no group of automorphisms can act transitively on its 99 vertices.
• 1988 — Brouwer and Neumaier, A remark on partial linear spaces of girth 5 with an application to strongly regular graphs, Combinatorica 8, 57–61.
• 2004 — Makhnev and Minakova, On automorphisms of strongly regular graphs with $\lambda=1$, $\mu=2$, Discrete Math. Appl. 14(2), and 2011 — Behbahani and Lam, Strongly regular graphs with non-trivial automorphisms, Discrete Math. 311, 132–144: further restrictions on the possible automorphism groups.
• 2014/2017 — Conway's prize offer publicises the problem.

No graph with these parameters has been found, and no non-existence proof is known.

Setting

Fix a finite vertex set $V$ and a simple graph $g$ on $V$ (irreflexive, symmetric adjacency $\mathrm{Adj}$). For vertices $v,w$ write $N(v) = \{u : \mathrm{Adj}(v,u)\}$ for the neighbourhood of $v$ and $N(v)\cap N(w)$ for the set of common neighbours. The graph $g$ is strongly regular with parameters $(n,k,\lambda,\mu)$, written $\mathrm{IsSRGWith}\ g\ n\ k\ \lambda\ \mu$, when

• $|V| = n$;
• $|N(v)| = k$ for every vertex $v$;
• $|N(v)\cap N(w)| = \lambda$ whenever $v$ and $w$ are adjacent;
• $|N(v)\cap N(w)| = \mu$ whenever $v \neq w$ are non-adjacent.

The case $\lambda = 1$ says that every edge lies in exactly one triangle — equivalently, the neighbourhood of each vertex induces a perfect matching, so such graphs are locally linear. The case $\mu = 2$ says that every non-adjacent pair is the pair of opposite corners of exactly one $4$-cycle. Conway's problem asks for $(n,k) = (99,14)$ with these two local conditions.

Counting paths of length two from a fixed vertex gives $k(k-\lambda-1) = (n-k-1)\mu$, which for $\lambda=1$, $\mu=2$ reduces to $2n = k^2 + 2$; with $k = 14$ this yields $n = 99$. Writing $A$ for the adjacency matrix, $I$ for the identity and $J$ for the all-ones matrix, strong regularity is equivalent to the matrix identity $A^2 = kI + \lambda A + \mu(J - I - A)$, which for $(99,14,1,2)$ reads $A^2 + A = 12I + 2J$; the eigenvalues of $A$ other than $k=14$ are then $3$ and $-4$, and integrality of their multiplicities ($54$ and $44$) is one of the feasibility conditions that $(99,14,1,2)$ passes.

Formalization targets

Goal

$$\exists\ \alpha,\ \exists\ g \text{ a simple graph on } \alpha,\quad \mathrm{IsSRGWith}\ g\ 99\ 14\ 1\ 2 .$$

The goal is Mathlib's own proof_wanted conway_99 in Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean, stated verbatim: existence of a finite type carrying a strongly regular graph with parameters $(99,14,1,2)$. A resolution in either direction is welcome — a proof settles the existence half, and a proof of the negation settles the non-existence half; the platform records the two as proof and disproof of the same statement.

Supporting targets

$$2n = k^2 + 2, \qquad k \text{ even}, \qquad k \in \{2,4,14,22,112,994\}$$

for every strongly regular graph with $\lambda = 1$, $\mu = 2$: the counting identity, local linearity, and the integrality restriction that cuts the family down to five non-degenerate parameter tuples.

$$\exists\, g,\ \mathrm{IsSRGWith}\ g\ 9\ 4\ 1\ 2, \qquad \exists\, g,\ \mathrm{IsSRGWith}\ g\ 243\ 22\ 1\ 2$$

the two members of the family that are known to exist: the $3\times 3$ rook's graph (the Paley graph on $9$ vertices) and the Berlekamp–van Lint–Seidel graph.

$$|E(g)| = 693, \qquad |\{\text{triangles of } g\}| = 231, \qquad A^2 + A = 12I + 2J, \qquad g \text{ not vertex-transitive}$$

structural consequences for a hypothetical $99$-graph, the last one being Wilbrink's theorem.

Significance

A $(99,14,1,2)$ graph, if it exists, is a locally linear graph of maximal density in its parameter range and a partial linear space of girth $5$ with $99$ points and $231$ lines of size $3$; its existence would also produce new association schemes and new examples for the general classification of strongly regular graphs. A non-existence proof would be the first case in this family ruled out by anything other than the classical feasibility conditions, and would say something new about how far local conditions ($\lambda=1$, $\mu=2$) constrain global structure.

Nothing in this mission is presently formalized. Mathlib defines SimpleGraph.IsSRGWith, proves the counting identity IsSRGWith.param_eq, the complement rule IsSRGWith.compl, and the matrix identity IsSRGWith.matrix_eq, and records the $99$-graph problem as a proof_wanted. The supporting targets are of three kinds: results that are proved in the literature and only need formalizing (existence at $(9,4,1,2)$ and $(243,22,1,2)$; Wilbrink's non-vertex-transitivity; the integrality restriction on $k$); routine consequences that supply reusable infrastructure (edge and triangle counts, the spectral identity, evenness of $k$); and the goal itself, which is open mathematics.

Difficulty

The obvious approaches fail for concrete reasons. Exhaustive search is out of range: the graph has $693$ edges among $\binom{99}{2} = 4851$ pairs, and no isomorph-free generation of locally linear graphs on $99$ vertices is feasible. Algebraic constructions are blocked by Wilbrink's theorem — the graph cannot be vertex-transitive, so it is not a Cayley graph and cannot be produced by the group-theoretic constructions that yield most known strongly regular graphs, including the two that work at $(9,4,1,2)$ and $(243,22,1,2)$. On the non-existence side, every classical feasibility test (the counting identity, integrality of the eigenvalue multiplicities, the Krein conditions, the absolute bound) is passed by $(99,14,1,2)$, so a proof of non-existence needs an argument that does not factor through the parameters alone.

Formalization scope

All statements are phrased with Mathlib's SimpleGraph.IsSRGWith on a Fintype vertex type with DecidableRel adjacency, and use Fintype.card, SimpleGraph.edgeFinset, SimpleGraph.cliqueFinset 3 (triangles as $3$-cliques), SimpleGraph.adjMatrix over $\mathbb{Z}$, and graph isomorphisms g ≃g g for automorphisms. The goal quantifies over α : Type together with a Fintype α instance, so the vertex set is finite by construction and the empty type does not satisfy the cardinality clause; the statement is therefore not vacuously satisfiable. Note that Mathlib's definition constrains $\lambda$ only through pairs that are actually adjacent and $\mu$ only through pairs that are actually distinct and non-adjacent, so degenerate small graphs (the one-vertex graph, $K_3$) do satisfy IsSRGWith with $\lambda = 1$, $\mu = 2$; the supporting statements carry the cardinality hypotheses ($0 < n$, $1 < n$) that exclude them where needed, and the degenerate degree $k = 2$ is listed explicitly in the classification of feasible degrees.

Infrastructure a complete development needs, and which is reusable beyond this mission: interface lemmas for counting common neighbours in a strongly regular graph; the spectral theory of the adjacency matrix (multiplicities of the two non-principal eigenvalues, and their integrality), which is the missing ingredient for the classification of feasible degrees; a Lean construction of the perfect ternary Golay code and its coset graph, for the $(243,22,1,2)$ case; and decision procedures for strong regularity of an explicitly given small graph, for the $(9,4,1,2)$ case. Contributions to any of these are welcome, as are partial non-existence results (for instance, restrictions on automorphisms of prime order) submitted as separate statements.

Selected references

• N. Biggs, Finite Groups of Automorphisms: Course Given at the University of Southampton, October–December 1969, London Mathematical Society Lecture Note Series 6, Cambridge University Press, 1971, p. 111.
• E. R. Berlekamp, J. H. van Lint, J. J. Seidel, A strongly regular graph derived from the perfect ternary Golay code, in: A Survey of Combinatorial Theory, North-Holland, 1973, pp. 25–30.
• R. K. Guy, Problems, in: The Geometry of Metric and Linear Spaces, Springer Lecture Notes in Mathematics 490, 1975, pp. 233–244 (Problem 7, J. J. Seidel, pp. 237–238). doi:10.1007/BFb0081147
• H. A. Wilbrink, On the $(99,14,1,2)$ strongly regular graph, in: Papers dedicated to J. J. Seidel, EUT Report 84-WSK-03, Eindhoven University of Technology, 1984, pp. 342–355. PDF
• A. E. Brouwer, A. Neumaier, A remark on partial linear spaces of girth 5 with an application to strongly regular graphs, Combinatorica 8 (1988), 57–61. doi:10.1007/BF02122552
• A. A. Makhnev, I. M. Minakova, On automorphisms of strongly regular graphs with $\lambda=1$, $\mu=2$, Discrete Mathematics and Applications 14 (2004), no. 2. doi:10.1515/156939204872374
• M. Behbahani, C. Lam, Strongly regular graphs with non-trivial automorphisms, Discrete Mathematics 311 (2011), 132–144. doi:10.1016/j.disc.2010.10.005
• J. H. Conway, Five $1,000 Problems (Update 2017), OEIS. PDF