Subreciprocal function and ell divisibility

A function $\mathrm{\ell }:(0,1\mathrm{/}2)\to (0,\mathrm{\infty })$ is subreciprocal when it is nonincreasing and satisfies $1<\mathrm{\ell }(x)\le 1\mathrm{/}x$ throughout its domain.

A nonempty finite graph $H$ is $\mathrm{\ell }$-divisive if there are witnesses $0<c<1\mathrm{/}2$ and $d>1$ such that for every $0<x<c$ and every nonempty finite graph $G$, the inequality ${\mathrm{ind}}_H(G)\le x^d\mathrm{\mid }G{\mathrm{\mid }}^{\mathrm{\mid }H\mathrm{\mid }}$ implies a QID block sequence of length at least $\mathrm{\ell }(x)$, width at least $\lfloor x^d\mathrm{\mid }G\mathrm{\mid }\rfloor$, uniformly $x$-sparse in one of $G,\bar{G}$.

Counts use Induced copy density and homogeneous restriction parameter, and empty blocks have the convention of Qid restricted blockade with empty blocks. Floors mean Integer part: for every real $x$ there is exactly one integer $m$ with $m\le x<m+1$. Logarithmic functions used here have base two, in the sense of The logarithm to a positive base other than one. The witnesses are fixed for $H,\mathrm{\ell }$, independently of $x,G$.

Proof/convention locator: Bucic, Nguyen, Scott and Seymour, Induced subgraph density I, Section 5 before 5.1.