Open problems and where to dig¶

What is not known: the undecided terms, conjectures supported by the data, and related questions.

1. Twelve lines: settled¶

\(c(12) = 336\). The last two types were decided in opposite directions: \(\{3^{18}\}\) is realizable with small integer coefficients (Example C3c), and \(\{4^5, 3^9\}\) is not even orientable (Proposition C3b). Two types, \(\{6, 4, 3^{12}\}\) and \(\{6, 3^{14}\}\), are proved non-realizable although they are orientable: the pencil types of 12 lines and of 12 pseudolines differ (Theorems C4, C5).

How the orientable cases were settled: for a single linear space, non-realizability is a statement about the solutions of a polynomial system; we prove it by exhibiting non-collinear triples whose determinants' product lies in the radical of the incidence ideal (a Gröbner-basis certificate, see the algorithms page); an incidence theorem (like Pappus for the non-Pappus configuration) is the human-readable special case. For a type, every orientable linear space of that type must be treated, which requires the engine's complete enumeration (mode 5) and an exact isomorphism reduction. The outcome for \(\{6, 3^{14}\}\) and \(\{6, 4, 3^{12}\}\) is recorded on the theorems page.

2. Thirteen lines¶

\(620 \le c(13) \le 659\): 620 types have certificates and 39 are undecided. The constructive searches of algorithms page (zone walk, targeted deformation, mirror sections) found no numerical solution for any of the 39, which suggests that most of them are not realizable. Deciding them needs the orientability search and, where it says "orientable", the algebraic method used at \(n = 12\); several of these types are harder than anything at \(n = 12\).

3. How fast do b(n) and c(n) grow?¶

Known: \(\log a(n) \asymp n^{2/3}\) (partitions into triangular numbers) and \(\log c(n) \gtrsim n^{1/2}\) (pencils that share no lines realize every partition of \(m \le n\) into parts \(\ge 3\)). The ratios \(c(n)/a(n)\) for \(n = 6, \dots, 12\) are \(0.60, 0.50, 0.42, 0.38, 0.33, 0.29, 0.26\): geometry eats a growing share of the arithmetic candidates, but slowly. A better lower bound would come from a construction that realizes many types at once; a better upper bound from a structural restriction on how many large pencils can coexist (Melchior's inequality is such a restriction, but it is far from tight for \(k = 3\)).

4. The intersection-point sequence beyond n = 13¶

Our bounds (spoke, Melchior, Hirzebruch) show that an arrangement of 14 lines with 27 or 28 points would have a largest pencil of five to seven lines while its other lines meet in very few points, a very dense configuration. We expect that it does not exist, which would give \(d(14) = 65\), but we have no proof and no search that covers it.

The precise threshold \(n^*\) from which the Salamon–Erdős description of \(V(n)\) is exact is open (they expect it to be small). Theorem E3 shows that one feature of it, that \(2n-2\) is the fourth smallest value, holds from \(n = 12\) on (for \(n = 12\) by computation).

5. Fields of definition¶

For \(n \le 14\) all our certificates are over \(\mathbb Q\) or \(\mathbb Q(\sqrt d)\), \(d \in \{2, 3, 5\}\), but except for the pentagram we never proved that the irrationality is needed; most of the \(\sqrt d\) examples probably have rational cousins. Rational realizability is a different sieve ("drawable on graph paper") that would give yet another sequence.

6. Other geometries¶

The same question makes sense elsewhere and we have not touched it:

• Pseudolines. Replace lines by curves that pairwise cross exactly once. Orientability becomes the exact criterion (Folkman–Lawrence), so the sequence is computable in principle; it agrees with \(c(n)\) for \(n \le 11\) and exceeds it at \(n = 12\) (problem 1).
• Complex lines. The Fano type is still impossible, but Möbius–Kantor (\(n = 8\)) and Hesse (\(n = 9\)) become possible: the complex sequence is \(1, 1, 2, 3, 5, 9, 16, 29, \ge 54, \dots\)
• Lines over finite fields; the sphere (great circles, where every pencil doubles); the hyperbolic plane (where non-intersecting lines appear and \(f_0\) can be anything from \(0\) to \(\binom n2\)).

7. Pencil types and the orchard problem¶

\(c(n)\) contains the orchard problem: the maximal \(t_3\) for \(n\) lines is the largest \(t\) with \(\{3^t\} \cup\) (anything) realizable. Conversely the orchard literature gives the only non-trivial upper bounds we know for a single pencil size (A003035: \(19\) for \(n = 12\), \(22\) for \(n = 13\)). For \(n = 12\) we reproved the bound for the type \(\{3^{20}\}\) alone: its linear spaces are the Steiner triple systems of order 13 minus a point, and none of them is orientable (Proposition C3a).