The price of homogeneity is polynomial

We provide explicit and polynomial bounds for the Homogeneous Wall Lemma which occurred for the first time implicitly in the $13$th entry of Robertson and Seymour’s Graph Minors Series [JCTB 1990] and has since become a cornerstone in the algorithmic theory of graph minors. A wall where each brick is assigned a set of colours is said to be homogeneous if each brick is assigned the same set of colours. The Homogeneous Wall Lemma says that there exists a function $h$ that, given non-negative integers $q$ and $k$ and an $h(q,k)$-wall $W$ where each brick is assigned a, possibly empty, subset of $\{1,\dots ,q\}$ contains a $k$-wall $W^{\mathrm{\prime }}$ as a subgraph such that, if one assigns to each brick $B$ of $W^{\mathrm{\prime }}$ the union of the sets assigned to the bricks of $W$ in its interior, then $W^{\mathrm{\prime }}$ is homogeneous. It is well-known that $h(q,k)\in k^{\mathcal{O}(q)}$. The Homogeneous Wall Lemma plays a key role in most applications of the Irrelevant Vertex Technique where an exponential dependency of $h$ on $q$ usually causes non-uniform dependencies on meta-parameters at best and additional exponential blow-ups at worst. By proving that $h(q,k)\in \mathcal{O}(q^4\cdot k^6)$, we provide a positive answer to a problem raised by Sau, Stamoulis, and Thilikos [ICALP 2020].