Curves, points, incidences and covering

TL;DR

This paper investigates geometric covering numbers: the minimum number of curves of a given type needed to cover a finite point set, with a focus on planar grids. It develops multiple exact or near-tight results across line covers, monotone curves, and several closed-curve families, employing tools from combinatorics, incidence geometry, and the Combinatorial Nullstellensatz. Key findings include exact grid-cover counts for lines, a Dilworth-based characterization for monotone curves, and bounds for circles, convex, and orthoconvex curves, along with a counterexample to a converse-covering conjecture. The work advances understanding of how different geometric structures interact with regular point configurations and suggests multiple open directions, including higher-dimensional skew coverings and tighter orthoconvex bounds.

Abstract

Given a point set, mostly a grid in our case, we seek upper and lower bounds on the number of curves that are needed to cover the point set. We say a curve covers a point if the curve passes through the point. We consider such coverings by monotonic curves, lines, orthoconvex curves, circles, etc. We also study a problem that is converse of the covering problem -- if a set of $n^2$ points in the plane is covered by $n$ lines then can we say something about the configuration of the points?

Figures (4)

• Figure 1: An othoconvex curve and its inner corners (in red)
• Figure 2: Orthoconvex curves with at most one inner corner
• Figure 3: Covering of $5\times 5$ grid by two orthoconvex curves (with at most one inner corner)
• Figure 4: Decomposition of orthoconvex curves with 2 inner corners

Theorems & Definitions (36)

• Definition 1
• Proposition 2
• proof
• Remark 3: Skew lines
• Theorem 4
• proof
• Remark 5
• Lemma 6: Combinatorial Nullstellensatz alon_1999
• Theorem 7
• proof