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The largest $K_r$-free set of vertices in a random graph

Tom Bohman, Marcus Michelen, Dhruv Mubayi

Abstract

For $r \ge 2$ and a graph $G$, let $α_{r}(G)$ be the maximum number of vertices in a $K_r$-free subgraph of $G$. We investigate the value $α_{r}(G)$ when $G$ is the random graph $G \sim G_{n, 1/2}$ and discover the following phenomenon: with high probability, $α_r(G)$ lies in an interval of constant length that varies in a non-monotonic fashion from $1$ to $\lfloor r/2\rfloor+1$ depending on the value of $n$. The special case $r=2$ corresponds to the independence number of random graphs which is well-known to have two-point concentration; our results therefore extend and generalize this basic fact in random graph theory, showing more complicated behavior when $r>2$. We also prove similar results where $K_r$ is replaced by any color critical graph like $C_5$.

The largest $K_r$-free set of vertices in a random graph

Abstract

For and a graph , let be the maximum number of vertices in a -free subgraph of . We investigate the value when is the random graph and discover the following phenomenon: with high probability, lies in an interval of constant length that varies in a non-monotonic fashion from to depending on the value of . The special case corresponds to the independence number of random graphs which is well-known to have two-point concentration; our results therefore extend and generalize this basic fact in random graph theory, showing more complicated behavior when . We also prove similar results where is replaced by any color critical graph like .
Paper Structure (14 sections, 22 theorems, 120 equations, 2 figures)

This paper contains 14 sections, 22 theorems, 120 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The case of triangles and a heuristic description for larger cliques
  3. Concentration via Poissonian statistics
  4. The interval of concentration for Kr+1
  5. Color-critical graphs and other properties
  6. Preparations for proof of Theorem 1
  7. Useful asymptotics
  8. Poisson Approximation
  9. Janson's inequality
  10. The lower bound: covering defects with independent sets
  11. The upper bound: not covering too many defects
  12. Proof of Theorem \ref{['thm:color-critical-concentration']}
  13. Concluding Remarks
  14. Proof of Proposition 18

Key Result

Theorem 1

Let $k = 2 \log_2 n + O(\log\log n)$, $r = o(\log \log n/\log\log\log n)$. Then, for $n$ sufficiently large and $j \in \{1, \dots, r\}$ we have For $\lambda=\mathbb{E} [Z_{k+1,\mu_{j}}] \leq n^{1/4}$ we also have

Figures (2)

  • Figure 1: These images depict likely maximum induced subgraphs of $G_{n,1/2}$ not containing $K_{12}$ at different points in the evolution of $G_{n,1/2}$ (where we view $n$ as growing and the edge probability is fixed at $1/2$). The black circles represent sets of $k+1$ vertices that have some number of defects, which are the gray edges. The gray ovals are independent sets of size $k$, and the dotted lines indicate which $k$-sets cover which defects. Note that we have $\mu_1=10, \xi_1=1; \mu_2= 4, \xi_2=1; \mu_3=2, \xi_3=1; \mu_4=1, \xi_4=1; \mu_{5}=1, \xi_5=4$ and ${\mathcal{J}} = \{1,2,3,5,11\}$. We placed a $\mathcal{J}$ for sets whose size lies in $\mathcal{J} \mod 11$. These sizes form the endpoints of the intervals of concentration of $X$.
  • Figure 2: Two structures that could achieve $X \ge rk+3$ in the case $r=11$ (where we seek a $K_{12}$-free set) are shown. Recall that $\mu_3=2, \xi_3=1$, and whp $Z_{k+1, 2} \ge 1$ implies that the standard structure, which is the structure in the first row, appears whp. However, if $Z_{k+1,2} \ge 3$ we could also achieve $X \ge kr+3$ with the second structure depicted here. In this second structure the gray rectangle is an independent set of size $k-1$ that covers the three defects in the fourth $(k+1)$-set.

Theorems & Definitions (47)

  • Theorem 1
  • Theorem 2
  • proof
  • Theorem 3
  • Remark 4
  • proof
  • proof
  • Theorem 8
  • Lemma 9
  • proof
  • ...and 37 more