Property (*) and Comb Outcomes — Examples

The four sets $B_1,B_2,B_3,B_4$ are pairwise disjoint and each has $4$ vertices, so $(B_1,B_2,B_3,B_4)$ is a $(4,4)$-blockade.

Each tooth $a_i$ is adjacent to every vertex of $B_i$ and to no vertex of $B_j$ for $j\mathrm{\ne }i$, and the teeth $a_1,a_2,a_3,a_4$ are distinct and lie outside the blocks. Hence $((a_i,B_i):i\in [4])$ is a $(4,4)$-comb by [L1].

The vertex $v$ lies outside the comb, is adjacent to every vertex in ${\bigcup }_{i=1}^4{B}_i$, and is nonadjacent to every $a_i$. Since this comb has $\mathrm{\ell }=w=4$, the pair consisting of the comb and $v$ realizes the geometric trigger configuration occurring in [L2]. No assertion that an unspecified family has property $(\ast )$ is being made.

Applying [L1] with $\mathrm{\ell }=4$ yields a pure blockade whose width is $\frac{w}{{\mathrm{\ell }}^2}=\frac{w}{16}\mathrm{.}$

Since [L1] names the blockade pure rather than complete or anticomplete, this branch keeps exactly the distinction used later on the A page: every pair of blocks is pure, but no stronger global uniformity is asserted.

Since $k=\sqrt{\mathrm{\ell }}$, one has ${\mathrm{\ell }}^{29c_4}=(k^2{)}^{29c_4}=k^{58c_4}=k^d\mathrm{.}$

Therefore $\frac{\mathrm{\mid }F\mathrm{\mid }}{{\mathrm{\ell }}^{29c_4}}=\frac{\mathrm{\mid }F\mathrm{\mid }}{k^d}\ge \frac{\mathrm{\mid }G\mathrm{\mid }}{k^d},$ using the assumption $\mathrm{\mid }F\mathrm{\mid }\ge \mathrm{\mid }G\mathrm{\mid }$.

This standalone calculation isolates the square-root renormalization: after replacing $\mathrm{\ell }$ by $k=\sqrt{\mathrm{\ell }}$, the width bound takes exactly the target form $\mathrm{\mid }G\mathrm{\mid }\mathrm{/}{k}^d$ when the numerical relation $d=58c_4$ holds.

The defining substitution gives $x={ϵ}^{5d}=2^{-10d}<2^{-d}\mathrm{.}$ Together with $d=58c_4^{\mathrm{\prime }}$, this verifies the two parameter inequalities required before applying the Rödl-initialized theorem; its separate graph-order hypothesis must also be checked in any application.

For the restricted-set branch, $x^{23c_4^{\mathrm{\prime }}}{ϵ}^d={ϵ}^{115c_4^{\mathrm{\prime }}d+d}\ge {ϵ}^{116c_4^{\mathrm{\prime }}d},$ because $115c_4^{\mathrm{\prime }}+1\le 116c_4^{\mathrm{\prime }}$ when $c_4^{\mathrm{\prime }}=4$.

For the clique-or-stable-set branch, $x^{31c_4^{\mathrm{\prime }}}{ϵ}^d={ϵ}^{155c_4^{\mathrm{\prime }}d+d}\ge {ϵ}^{156c_4^{\mathrm{\prime }}d},$ because $155c_4^{\mathrm{\prime }}+1\le 156c_4^{\mathrm{\prime }}$.

If $k\ge x^{-1}$, then ${ϵ}^d=x^{1\mathrm{/}5}\ge k^{-1\mathrm{/}5},$ so $\frac{{ϵ}^d}{k^d}\ge \frac{1}{k^{2d}}\mathrm{.}$ This is exactly the comparison used to turn the complete-or-anticomplete blockade branch into the generalized-nice blockade outcome.

These computations are the concrete numerical version of the four exponent transfers behind the local blockade claim and the final proof of generalized niceness.