Let $G$ be a graph, $S \subseteq V(G)$ be a vertex set in $G$ and $r$ be a positive integer. The distance $r$-independence number of $S$ is the size of the largest subset $I \subseteq S$ such that no pair $u$, $v$ of vertices in $I$ have a path on at most $r$ edges between them in $G$. It has been conjectured [Chudnovsky et al., arXiv, 2025] that for every positive integer $t$ there exist positive integers $c$, $d$ such that every graph $G$ that excludes both the complete bipartite graph $K_{t,t}$ and the grid $\boxplus_t$ as an induced minor has a tree decomposition in which every bag has (distance $1$) independence number at most $c(\log n)^d$. We prove a weaker version of this conjecture where every bag of the tree decomposition has distance $16(\log n + 1)$-independence number at most $c(\log n)^d$. On the way we also prove a version of the conjecture where every bag of the decomposition has distance $8$-independence number at most $2^{c (\log n)^{1-(1/d)}}$.

翻译：设$G$为一个图，$S \subseteq V(G)$是$G$中的一个顶点子集，$r$为正整数。$S$的距离$r$独立数定义为最大子集$I \subseteq S$的大小，使得$I$中任意两个顶点$u$、$v$在$G$中不存在长度至多为$r$的路径相连。已有猜想[Chudnovsky et al., arXiv, 2025]指出：对任意正整数$t$，存在正整数$c$、$d$，使得每个禁止完全二分图$K_{t,t}$和网格图$\boxplus_t$作为诱导子式的图$G$，都存在一个树分解，其中每个袋的（距离$1$）独立数至多为$c(\log n)^d$。我们证明了该猜想的弱化版本：其中树分解每个袋的距离$16(\log n + 1)$独立数至多为$c(\log n)^d$。此外，我们还在推导过程中证明了该猜想的另一个版本：其中分解每个袋的距离$8$独立数至多为$2^{c (\log n)^{1-(1/d)}}$。