Excursions in Sylvester-Gallai land

The Sylvester-Gallai theorem states that for a finite set of points in the plane, if every line determined by any two of these points also contains a third, then the set is necessarily made of collinear points. In this paper, we first provide a counterexample in the plane when the point set is countably infinite but bounded. Then we consider a variant of the Sylvester-Gallai theorem where instead of a finite point set we have a finite family of convex sets in $\mathbb{R}^d$ ($d\geq 2$). Finally, we present another variant of the Sylvester-Gallai theorem, when instead of point sets we have a finite family of line-segments in the plane.

💡 Research Summary

The paper “Excursions in Sylvester‑Gallai land” investigates three distinct extensions of the classical Sylvester‑Gallai theorem, which states that a finite set of points in the Euclidean plane in which every line determined by two points contains a third must be collinear. The authors first ask whether a similar statement holds for infinite point sets, then replace points by convex bodies, and finally replace points by line‑segments.

1. A bounded countable counterexample.
The authors construct a bounded, countably infinite set H⊂ℝ² with exactly one accumulation point (the origin) that satisfies the Sylvester‑Gallai condition: any line through two distinct points of H contains a third point of H. The construction uses six half‑rays L₁,…,L₆ emanating from the origin at 60° intervals. On the odd‑indexed rays they place points at distances 1/(3j‑2) (j∈ℕ); on the even‑indexed rays they place points at distances 1/(3j‑1). Lemma 2.1 gives a simple formula for the length of the angle bisector in a triangle with a 2π/3 angle, namely 1/b = 1/x + 1/y, derived from an elementary area computation. Using this lemma, the authors show that for any two distinct points p,q∈H the line pq either passes through the origin (hence contains a third point) or makes an angle of π/3 or 2π/3 with the origin; in each case a third point on the same line can be explicitly constructed by choosing an appropriate index. Thus H is a minimal (one‑accumulation‑point) counterexample to any naïve infinite‑set extension of Sylvester‑Gallai.

2. Convex systems.
A “convex system” is defined as a finite family F={K₁,…,Kₙ} of pairwise disjoint, compact, strictly convex sets in ℝᵈ (d≥2), each with non‑empty interior. Let U = ⋃Kᵢ and call a line ordinary if it meets U in exactly two points (necessarily belonging to two different Kᵢ).

Dimension 2. Theorem 3.1 states that a convex system in the plane has no ordinary line if and only if n>3. For n=2 the four common external tangents are ordinary. For n=3 the authors analyse the boundary of the convex hull C = conv U and show that unless a very specific configuration occurs (illustrated in Figure 3.1), an ordinary line always exists. When n>3 they give an explicit geometric construction (Figure 3.2) where K₁ and K₃ are convex quadrilaterals, K₂ and K₄ are “almost‑concave” polygons whose edges are large‑radius circular arcs, and the remaining n‑4 sets are placed in the middle region. By arranging the sets so that every potential two‑point line is intercepted by a third set, they obtain a convex system with no ordinary line.

Higher dimensions (d≥3). Theorem 4.1 proves that any convex system with n≥2 in ℝᵈ (d≥3) always possesses an ordinary line. The proof proceeds by intersecting the convex hull C with a supporting hyperplane P at a boundary point z∉U. The intersection P∩Kᵢ is either empty or a single point aᵢ. If exactly two such points appear, the line a₁a₂ is ordinary. If more than two appear, the classical Sylvester‑Gallai theorem applied to {a₁,…,a_m} yields an ordinary line unless all aᵢ are collinear. In the collinear case the authors tilt the supporting hyperplane slightly (introducing a small ε‑perturbation) to obtain a new plane P* that meets only two of the original sets, reducing the problem to the planar case. For d>3 the same argument works after reducing to a (d‑1)‑dimensional Sylvester‑Gallai theorem (or by selecting four interior points and working in the 3‑dimensional affine span).

Thus the paper completely classifies the existence of ordinary lines for convex systems: in the plane the threshold is n>3, while in any higher dimension ordinary lines are unavoidable.

3. Families of line‑segments.
The final section studies finite families F of line‑segments in ℝ² with pairwise disjoint relative interiors.

Theorem 5.1 asserts that if |F|≤5, the union S_F is not collinear, and no line meets the larger set M⊇S_F in exactly two points (M\S_F is finite), then M must be collinear. The proof proceeds by considering the convex hull P = conv S_F. For each vertex a of P the authors show that at least three segments of F must have an endpoint at a; otherwise one can find a line intersecting M in infinitely many points, contradicting the finiteness of M\S_F. This forces |F|≥6, a contradiction. Hence P cannot be a polygon, and S_F must lie on a line. The bound “5” is shown to be sharp by exhibiting counterexamples with six segments (Figure 5.1).

Theorem 5.3 refines the six‑segment case: if |F|=6 and the geometric graph formed by S_F is 3‑connected, then the configuration must be one of four combinatorial types shown in Figure 5.1 (a)–(d). The proof first shows that P must be a triangle; the remaining three segments A, B, C each attach to a distinct vertex. 3‑connectivity forces each vertex to have degree at least three, which restricts how A, B, C can be placed, leading precisely to the four illustrated possibilities.

The authors also discuss that for |F|=7 many more 3‑connected graphs arise (Figure 5.2).

Overall assessment.
The paper makes three substantive contributions: (i) a minimal bounded countable counterexample to an infinite‑set Sylvester‑Gallai statement, (ii) a complete dichotomy for ordinary lines in convex systems across dimensions, and (iii) tight combinatorial bounds for families of line‑segments. The constructions are elegant, especially the planar convex system without ordinary lines for n>3, and the higher‑dimensional ordinary‑line proof cleverly reduces to lower‑dimensional cases using supporting hyperplanes. The line‑segment results blend geometric reasoning with graph‑theoretic connectivity, yielding a clear classification.

Some minor issues remain. The figures (especially Figures 3.2 and 4.1) are described only verbally; providing explicit coordinates or analytic descriptions would aid reproducibility. Lemma 2.1’s statement about the “angle bisector length” could be clarified, as the notation b is introduced without definition. A few typographical errors (e.g., “con vex” split across lines) distract from readability. Nonetheless, the logical flow is sound, the results are novel, and the paper broadens the scope of Sylvester‑Gallai type theorems in a meaningful way.