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Concentration of the hypergraph's weak independence number

Stepan Vakhrushev

TL;DR

This paper studies the weak independence number $\alpha(H(n,k,p))$ in the binomial random $k$-uniform hypergraph $H(n,k,p)$. It generalizes the two-point concentration phenomenon from graphs to hypergraphs, proving concentration on two values for $p \ge n^{-{(k-1)k}/{(k+1)}+\varepsilon}$. The authors adapt the second moment method with augmented independent sets, introducing the parameters $s_x$, $s_z$, and $r_z$ to tightly locate the concentration window and provide a near-optimal anti-concentration regime. As a corollary, they obtain a precise asymptotic description via the inverse function $f_k$ and discuss extensions to more general $j$-independence numbers, highlighting the sharp threshold behavior and its implications for the independence structure of sparse random hypergraphs.

Abstract

In this note we generalize the results of the recent work by Tom Bohman and Jacob Hofstad on the independence number in G(n, p) to the case of the random k-uniform hypergraph. Concentration in two values occurs in the regime $p>n^{-(k-1)k/(k+1)+\varepsilon}$.

Concentration of the hypergraph's weak independence number

TL;DR

This paper studies the weak independence number in the binomial random -uniform hypergraph . It generalizes the two-point concentration phenomenon from graphs to hypergraphs, proving concentration on two values for . The authors adapt the second moment method with augmented independent sets, introducing the parameters , , and to tightly locate the concentration window and provide a near-optimal anti-concentration regime. As a corollary, they obtain a precise asymptotic description via the inverse function and discuss extensions to more general -independence numbers, highlighting the sharp threshold behavior and its implications for the independence structure of sparse random hypergraphs.

Abstract

In this note we generalize the results of the recent work by Tom Bohman and Jacob Hofstad on the independence number in G(n, p) to the case of the random k-uniform hypergraph. Concentration in two values occurs in the regime .
Paper Structure (22 sections, 16 theorems, 120 equations)

This paper contains 22 sections, 16 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Necessary Definitions from Hypergraph Theory
  3. Background
  4. New results
  5. Statement of the main theorems
  6. Proof technology
  7. Structure of the Proof of Theorem \ref{['th:main_res']}
  8. Preliminary calculations
  9. Binomial coefficients relations
  10. Asymptotics for values of $s=s(n)$ that are of the form \ref{['main_fm']}
  11. Defining and first estimating $s_z$ and $r_z$
  12. Estimating of $r_z$
  13. Estimating of $s_z$
  14. Upper bound
  15. Lower bound: second moment of $Z$
  16. ...and 7 more sections

Key Result

Theorem 1

Fix $k$ and $1 \leq j \leq k-1$ and let $p=\omega\left(n^{1-k}\right)$ and simultaneously $p=o\left(n^{j+1-k}\right)$. Then a.a.s. and, in particular,

Theorems & Definitions (29)

  • Theorem 1: (Consequence of sudakov)
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 5
  • Lemma 6
  • proof
  • Theorem 7
  • Lemma 8
  • proof
  • ...and 19 more