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Some exact values of the inducibility and statistics constants for hypercubes

Levente Bodnár, Oleg Pikhurko

TL;DR

The paper investigates inducibility and statistics densities of $d$-subcubes in the $n$-dimensional hypercube, using flag-algebra techniques to determine asymptotic limits for new cases. It obtains exact inducibility constants for five configurations $H\subseteq {0,1}^3$ and for the pairs $(d,s)=(3,2),(4,2),(4,4)$, with lower bounds derived from blowups of small Hamming codes and upper bounds certified by flag-algebra certificates. In particular, it proves $\\lambda(W_{12})=1/2$ and provides the full inducibility function for $W_{12}$, showing $\\Lambda(W_{12},n)=\tfrac{1}{2}{n\choose 3}2^{n-3}$ for $n\ge4$; similar precise values are obtained for the statistics constants $\\lambda(3,2)=8/9$, $\\lambda(4,2)=264/343$, and $\\lambda(4,4)=26/27$. The authors supply computational tools (SageMath-based certificates and notebooks) to verify the results and discuss open questions, including unresolved $W_3,W_4,W_5$ cases and related $\\lambda(d,s)$ values, illustrating the power and limitations of the flag-algebra approach in hypercube extremal problems.

Abstract

We consider two types of problems: maximising, over subsets $S\subseteq \{0,1\}^n$, the density of $d$-subcubes $C$ in the $n$-hypercube graph that span a subgraph such that $S\cap C$ is i) isomorphic to the given configuration $H\subseteq\{0,1\}^d$ (the inducibility problem), or ii) has the given size $s$ (the statistics problem). Using flag algebras, we determine the limit of this density as $n\to\infty$ for 5 new configurations $H\subseteq\{0,1\}^3$ and for 3 new pairs $(d,s)$, namely for $(3,2)$, $(4,2)$ and $(4,4)$. Interestingly, the lower bounds in the last three cases come from blowups of small Hamming codes.

Some exact values of the inducibility and statistics constants for hypercubes

TL;DR

The paper investigates inducibility and statistics densities of -subcubes in the -dimensional hypercube, using flag-algebra techniques to determine asymptotic limits for new cases. It obtains exact inducibility constants for five configurations and for the pairs , with lower bounds derived from blowups of small Hamming codes and upper bounds certified by flag-algebra certificates. In particular, it proves and provides the full inducibility function for , showing for ; similar precise values are obtained for the statistics constants , , and . The authors supply computational tools (SageMath-based certificates and notebooks) to verify the results and discuss open questions, including unresolved cases and related values, illustrating the power and limitations of the flag-algebra approach in hypercube extremal problems.

Abstract

We consider two types of problems: maximising, over subsets , the density of -subcubes in the -hypercube graph that span a subgraph such that is i) isomorphic to the given configuration (the inducibility problem), or ii) has the given size (the statistics problem). Using flag algebras, we determine the limit of this density as for 5 new configurations and for 3 new pairs , namely for , and . Interestingly, the lower bounds in the last three cases come from blowups of small Hamming codes.

Paper Structure

This paper contains 5 sections, 3 theorems, 18 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Lower bounds
  3. Upper bounds
  4. Inducibility function for $W_{12}$
  5. Concluding Remarks

Key Result

Theorem 1

We have the following inducibility constants: $\lambda(W_7)=1/3$, $\lambda(W_8)=2/3$, $\lambda(W_9)=4/9$, $\lambda(W_{10})=5/12$ and $\lambda(W_{12})=1/2$.

Figures (3)

  • Figure 1: Configurations $W_7, W_8, W_9, W_{10}, W_{12}$ respectively.
  • Figure 2: Unsolved configurations, $W_3, W_4$ and $W_5$.
  • Figure 3: The configurations with $c_H=0$. The top row shows $H_1, H_2, H_3$, the bottom row shows $H_4, H_5, H_6$.

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3