Many collinear k-tuples with no k+1 collinear points

For every $k>3$, we give a construction of planar point sets with many collinear $k$-tuples and no collinear $(k+1)$-tuples. We show that there are $n_0=n_0(k)$ and $c=c(k)$ such that if $n\geq n_0$, then there exists a set of $n$ points in the plane that does not contain $k+1$ points on a line, but it contains at least $n^{2-\frac{c}{\sqrt{\log n}}}$ collinear $k$-tuples of points. Thus, we significantly improve the previously best known lower bound for the largest number of collinear $k$-tuples in such a set, and get reasonably close to the trivial upper bound $O(n^2)$.

💡 Research Summary

The paper addresses a long‑standing question of Paul Erdős concerning point‑line incidences in the plane: for a fixed integer k > 3, how many k‑point collinear subsets can a planar set of n points contain if it is forbidden to have any (k + 1) points on a line? Formally, let t_k(P) be the number of lines meeting a point set P in exactly k points, and define t_k(n) as the maximum of t_k(P) over all n‑point sets with no (k + 1) collinear points. Erdős conjectured that t_k(n) = o(n²) for any fixed k > 3, offering a $100 prize for a proof or disproof.

Prior work gave only modest lower bounds: Kárteszi (1975) proved t_k(n) ≥ c_k n log n; Grünbaum (1976) improved this to t_k(n) ≥ c_k n^{1+1/(k‑2)}; later Brass, Ismailescu, and Elkies refined the exponent further, but the exponent still tended to 1 as k grew. Thus the gap between the trivial O(n²) upper bound and known lower bounds remained large.

The authors present a construction that pushes the lower bound much closer to the quadratic upper bound. Their main theorem states that for any integer k ≥ 4 there exist constants n₀(k) and c(k) = 2 log(4k + 9) such that for all n ≥ n₀, one can find an n‑point planar set with no (k + 1) collinear points and at least n^{2 − c/√log n} collinear k‑tuples. This exponent is of the form 2 − o(1), i.e., asymptotically almost quadratic.

The construction proceeds in several stages:

1. High‑dimensional lattice framework.
   The authors work initially in ℝ^d with d chosen large (depending on n). They consider the integer lattice ℤ^d and the closed Euclidean ball B_d(r) of radius r centered at the origin. Using classic volume‑approximation arguments (Gauss’s lattice point estimate) they establish Lemma 3, which bounds the number of lattice points N(B_d(r)) between the volumes of slightly smaller and larger balls. Lemma 4 gives an upper bound for the number of lattice points on a sphere S_d(r) in terms of the volume of a (d‑2)‑dimensional ball, multiplied by a factor O(log r log log r). These lemmas allow precise control of how many lattice points lie on a given sphere.

2. Even k construction.
   Choose r₀ = 2^d and consider all lattice points inside B_d(r₀). By the pigeon‑hole principle there exists a radius r ≤ r₀ such that a positive fraction (≥ 1/r₀²) of those points lie exactly on the sphere S_d(r). Among the points on S_d(r) consider unordered pairs (p,q). Their Euclidean distance squared is an integer bounded by (2r)², so there are at most 4r² possible distinct distances. Again by pigeon‑hole, many pairs share the same distance ℓ. Selecting one such pair (p₁,q₁) defines a line s. Along s we place k/2 − 1 additional points on each side of p₁ and q₁ at equal spacing ℓ, obtaining a total of k distinct integer lattice points. By construction each of these points lies on a sphere S_d(r_i) with r_i ≤ r(k + 1). Since the spheres are distinct, no line can contain more than k points from the whole set.

   Counting: the number of “good” pairs (same distance) is at least V(B_d(r₀ − √d/2)) / (8 r₀⁶). Each such pair yields a distinct k‑tuple line, so t_k(P) is bounded below by a constant times V(B_d(r₀ − √d/2)) / r₀⁶. Meanwhile the total number of points n is bounded above by a sum over the k/2 spheres, each contributing at most O(log r_i log log r_i V(B_{d‑2}(r_i))). Combining these estimates and using Stirling’s approximation for the volume of a d‑dimensional ball yields the inequality t_k(P) ≥ n^{2 − c/√log n}.

3. Odd k construction.
   The odd case is similar but requires an extra middle point to reach k points on a line. Starting again with a sphere S_d(r) containing many lattice points, the authors consider the scaled sphere S_d(2r) intersected with the even lattice (2ℤ)^d. For each point p on S_d(r) there is a corresponding point p′ on S_d(2r) with all coordinates even; the segment p′q′ has an integer midpoint m₀ ∈ ℤ^d. Using the same pigeon‑hole argument on distances (now up to 16r² possibilities) they find many pairs with the same distance 2ℓ. For each such pair they construct a line containing the two endpoints, the midpoint, and (k‑3)/2 equally spaced points on each side, again obtaining k distinct integer points. The points lie on a collection of (k‑1)/2 spheres together with a hyperplane α_{x₀} (the set of points whose first coordinate equals a fixed integer x₀). The hyperplane contains at most (k‑1)/2 spheres, guaranteeing that no line can host more than k points.

   The counting proceeds analogously to the even case, yielding the same asymptotic lower bound with the same constant c = 2 log(4k + 9).

4. Projection to the plane.
   After constructing the high‑dimensional set P ⊂ ℤ^d with the desired properties, the authors apply a generic linear projection π: ℝ^d → ℝ². By choosing the projection direction outside a finite union of “bad” subspaces, they ensure that π is injective on P and that collinearity relations are preserved: three non‑collinear points remain non‑collinear, and any line containing exactly k points of P projects to a line containing exactly k points of π(P). Consequently, π(P) is a planar point set of size n with no (k + 1) collinear points and at least the same number of k‑point collinear lines.

5. Comparison with previous work.
   The new bound n^{2 − c/√log n} dramatically improves upon the earlier best lower bounds of order n log n or n^{1+1/(k‑2)}. While still shy of the trivial O(n²) upper bound, the exponent now differs from 2 by only a term that tends to zero as n grows, showing that Erdős’s conjectured o(n²) behavior is essentially tight for fixed k.

In summary, the paper introduces a novel geometric‑combinatorial construction that leverages high‑dimensional lattice points on spheres, careful pigeon‑hole arguments on distances, and a projection step to obtain planar point sets with an almost quadratic number of k‑point collinear subsets while strictly forbidding any (k + 1) collinear points. This result narrows the gap between known lower and upper bounds and provides a new technique that may be useful for related incidence problems.