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Intersection patterns of set systems on manifolds with slowly growing homological shatter functions

Sergey Avvakumov, Marguerite Bin, Xavier Goaoc

Abstract

A theorem of Matoušek asserts that for any $k \ge 2$, any set system whose shatter function is $o(n^k)$ enjoys a fractional Helly theorem of order $k$: in the $k$-wise intersection hypergraph, positive density implies a linear-size clique. Kalai and Meshulam conjectured a generalization of that phenomenon to homological shatter functions. It was verified for set systems with bounded homological shatter functions and ground set with a forbidden homological minor (which includes $\mathbb{R}^d$ by a homological analogue of the van Kampen-Flores theorem). We present two contributions to this line of research: - We study homological minors in certain manifolds (possibly with boundary), for which we prove analogues of the van Kampen-Flores theorem and of the Hanani-Tutte theorem. - We introduce graded analogues of the Radon and Helly numbers of set systems and relate their growth rate to the original parameters. This allows to extend the verification of the Kalai-Meshulam conjecture for sufficiently slowly growing homological shatter functions.

Intersection patterns of set systems on manifolds with slowly growing homological shatter functions

Abstract

A theorem of Matoušek asserts that for any , any set system whose shatter function is enjoys a fractional Helly theorem of order : in the -wise intersection hypergraph, positive density implies a linear-size clique. Kalai and Meshulam conjectured a generalization of that phenomenon to homological shatter functions. It was verified for set systems with bounded homological shatter functions and ground set with a forbidden homological minor (which includes by a homological analogue of the van Kampen-Flores theorem). We present two contributions to this line of research: - We study homological minors in certain manifolds (possibly with boundary), for which we prove analogues of the van Kampen-Flores theorem and of the Hanani-Tutte theorem. - We introduce graded analogues of the Radon and Helly numbers of set systems and relate their growth rate to the original parameters. This allows to extend the verification of the Kalai-Meshulam conjecture for sufficiently slowly growing homological shatter functions.
Paper Structure (18 sections, 18 theorems, 11 equations, 7 figures)

This paper contains 18 sections, 18 theorems, 11 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Context and motivation
  3. Combinatorial background: convexity parameters of set systems
  4. Homological background: homological minors
  5. Parameters of set systems with a forbidden homological minor
  6. Proof of Conjecture \ref{['conj:kalai-meshulam']} for bounded homological shatter functions
  7. Statement of the results
  8. On homological minors
  9. A homological Hanani-Tutte theorem
  10. Forbidden homological minors for manifolds
  11. Graded parameters of set systems
  12. Graded Radon and Helly numbers
  13. Consequences for topological set systems
  14. Other graded parameters and relations
  15. Homological minors of manifolds and their triangulations
  16. ...and 3 more sections

Key Result

Theorem 2

For any $d \ge 1$, $\Delta_{d+2}^{(\lceil d/2 \rceil)}$ does not homologically almost embed into $\mathbb{R}^d$.

Figures (7)

  • Figure 1: The setup for the construction of $T'$ and $f'$.
  • Figure 2: The construction of $B'$.
  • Figure 3: The path $P_\sigma$.
  • Figure 4: The piping $F_\sigma$ between $D_{\sigma,x}$ and $D_{\sigma,y}$ in a neighborhood of $P_{\sigma}$.
  • Figure 5: The chain $C_\sigma = D_{\sigma,x} + \partial F_\sigma + D_{\sigma,y}$ over $\mathbb{Z}_2$.
  • ...and 2 more figures

Theorems & Definitions (31)

  • Conjecture 1: Kalai and Meshulam
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Corollary 6
  • Theorem 7
  • Lemma 2.1
  • Theorem 8
  • Lemma 3.1
  • ...and 21 more