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The classification of Boolean degree $1$ functions in high-dimensional finite vector spaces

Ferdinand Ihringer

TL;DR

This work classifies Boolean degree $1$ functions on the Grassmann scheme $J_q(n,k)$, equivalently Cameron–Liebler classes, in the regime $n \ge n_0(k,q)$. By leveraging Ramsey-type theorems for geometric lattices (GLR and Voigt) and a detailed weight-analysis, the authors reduce to the case $k=2$ and show that for sufficiently large $n$ all Boolean degree $1$ functions are trivial, thereby establishing the existence of a threshold $n_0(k,q)$. The results imply nonexistence of nontrivial two-intersection sets for large $n$ and yield improved lower bounds on the size of nontrivial functions in the small-dimension regime. The work connects Cameron–Liebler theory with blocking-set partitions (Mazzocca–Tallini) and opens avenues for extending the classification to higher degrees and related combinatorial geometries, with potential implications for design theory and complexity measures in theoretical computer science.

Abstract

We classify the Boolean degree $1$ functions of $k$-spaces in a vector space of dimension $n$ (also known as Cameron-Liebler classes) over the field with $q$ elements for $n \geq n_0(k, q)$. This also implies that two-intersecting sets with respect to $k$-spaces do not exist for $n \geq n_0(k, q)$. Our main ingredient is the Ramsey theory for geometric lattices.

The classification of Boolean degree $1$ functions in high-dimensional finite vector spaces

TL;DR

This work classifies Boolean degree functions on the Grassmann scheme , equivalently Cameron–Liebler classes, in the regime . By leveraging Ramsey-type theorems for geometric lattices (GLR and Voigt) and a detailed weight-analysis, the authors reduce to the case and show that for sufficiently large all Boolean degree functions are trivial, thereby establishing the existence of a threshold . The results imply nonexistence of nontrivial two-intersection sets for large and yield improved lower bounds on the size of nontrivial functions in the small-dimension regime. The work connects Cameron–Liebler theory with blocking-set partitions (Mazzocca–Tallini) and opens avenues for extending the classification to higher degrees and related combinatorial geometries, with potential implications for design theory and complexity measures in theoretical computer science.

Abstract

We classify the Boolean degree functions of -spaces in a vector space of dimension (also known as Cameron-Liebler classes) over the field with elements for . This also implies that two-intersecting sets with respect to -spaces do not exist for . Our main ingredient is the Ramsey theory for geometric lattices.
Paper Structure (7 sections, 16 theorems, 5 equations, 1 figure)

This paper contains 7 sections, 16 theorems, 5 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm:main']}
  4. Two-Intersection Sets
  5. Small Dimension
  6. On Mazzocca and Tallini
  7. Future Work

Key Result

Theorem 1.2

Let $q$ be a prime power and let $k$ be some integer, where $k \geq 2$. Then there exists a number $c_0(k, q)$ such that for all $\max(k, n-k) \geq c_0(k, q)$ all Boolean degree $1$ functions on $k$-spaces in an $n$-dimensional vector space over a finite field with $q$ elements are trivial.

Figures (1)

  • Figure 1: An illustration of the induction in the proof of Proposition \ref{['prop:rec_ram']} for $i=2$. The dotted lines correspond to $S_3^1$ and $S_3^2$, the thick line corresponds to $S_3^3$, and the dotted area corresponds to $V_3 = \langle P, S_3^3\rangle$.

Theorems & Definitions (26)

  • Theorem 1.2
  • Theorem 1.3: Graham, Leeb, Rothschild (1972)
  • Theorem 1.4: Graham, Leeb, Rothschild (1972)
  • Theorem 1.5: Voigt (1978)
  • Proposition 2.1
  • proof
  • Proposition 2.2: Drudge (1998)
  • Proposition 2.3: Metsch (2010)
  • Corollary 2.4
  • proof
  • ...and 16 more