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Covering points with planes

Hailong Dao, Manik Dhar, Izabella Łaba, Ben Lund

TL;DR

The paper investigates how large a finite set $S$ can be when every proper subset lies in a union of affine subspaces of prescribed dimensions, but $S$ itself does not, formalized via dimension vectors and covering configurations. It develops both algebraic (vanishing-ideal) and combinatorial methods to bound the maximal size $|S|$ through the invariant $C_A(V)$ and related notions, with tight results in several cases, particularly for identical-dimension plane configurations. It then extends the framework to Artinian rings and, in particular, to the ring $\mathbb{Z}/p^k\mathbb{Z}$, using multiscale p-adic geometry to obtain upper bounds in 2D and to construct large nearly-covered sets that defy naive field-based bounds. The work also connects to matroid theory by providing purely combinatorial bounds and discusses potential generalizations to non-representable matroids and higher-dimensional flats, highlighting the nuanced interplay between algebraic, combinatorial, and p-adic techniques.

Abstract

Suppose that each proper subset of a set $S$ of points in a vector space is contained in the union of planes of specified dimensions, but $S$ itself is not contained in any such union. How large can $|S|$ be? We prove a general upper bound on $|S|$, which is tight in some cases, for example when all of the planes have the same dimension. We produce an example showing that this upper bound does not hold for point sets whose proper subsets are covered by lines in $(\mathbb{Z}/p^k\mathbb{Z})^2$ with $k\geq 2$, and prove an upper bound in this case. We also investigate the analogous problem for general matroids.

Covering points with planes

TL;DR

The paper investigates how large a finite set $S$ can be when every proper subset lies in a union of affine subspaces of prescribed dimensions, but $S$ itself does not, formalized via dimension vectors and covering configurations. It develops both algebraic (vanishing-ideal) and combinatorial methods to bound the maximal size $|S|$ through the invariant $C_A(V)$ and related notions, with tight results in several cases, particularly for identical-dimension plane configurations. It then extends the framework to Artinian rings and, in particular, to the ring $\mathbb{Z}/p^k\mathbb{Z}$, using multiscale p-adic geometry to obtain upper bounds in 2D and to construct large nearly-covered sets that defy naive field-based bounds. The work also connects to matroid theory by providing purely combinatorial bounds and discusses potential generalizations to non-representable matroids and higher-dimensional flats, highlighting the nuanced interplay between algebraic, combinatorial, and p-adic techniques.

Abstract

Suppose that each proper subset of a set of points in a vector space is contained in the union of planes of specified dimensions, but itself is not contained in any such union. How large can be? We prove a general upper bound on , which is tight in some cases, for example when all of the planes have the same dimension. We produce an example showing that this upper bound does not hold for point sets whose proper subsets are covered by lines in with , and prove an upper bound in this case. We also investigate the analogous problem for general matroids.

Paper Structure

This paper contains 11 sections, 21 theorems, 38 equations, 4 figures.

Table of Contents

  1. Introduction
  2. General setup
  3. Organization
  4. Algebraic approach
  5. Generalized algebraic approach
  6. Combinatorial bounds
  7. Nearly covered sets in $(\mathbb{Z}/p^k\mathbb{Z})^n$
  8. Basic $p$-adic geometry
  9. Large nearly covered sets in $(\mathbb{Z}/p^k\mathbb{Z})^2$
  10. Upper bound for $p$-adic lines
  11. Acknowledgments

Key Result

theorem 1

For any $V$, there is a constant $C(V)$ such that if any subset of $S$ of size at most $C(V)$ satisfies $(D_V)$, then $S$ is $(D_V)$.

Figures (4)

  • Figure 1: Possible set of points and lines described in \ref{['ex:missingTriangle']} with $t=5$ and $s=1$. The point $P$ is blue, the additional points of $S$ are red, and the points of $T \setminus S$ are black. Four lines and two points suffice to cover $T \setminus \{P\}$.
  • Figure 2: Two sets of $6$ points with different underlying matroids that are each nearly covered by $2$ lines. The set of points on the right has the same underlying matroid as that described in \ref{['ex:triangle']}, and is obtained from the left set by removing the circuit denoted by the dashed line.
  • Figure 3: The flattened triangle with two points at each vertex, plus an additional point that requires an extra line.
  • Figure 4: If we remove one of the points in the triangle, then its companion and the extra point above can be covered by one line, and then one of the lines covering the triangle is no longer needed.

Theorems & Definitions (37)

  • theorem 1
  • theorem 2
  • theorem 3
  • theorem 4
  • lemma 1
  • proposition 1
  • proof
  • theorem 5
  • proof
  • theorem 6
  • ...and 27 more