Coherent graphs and support-regular blockades

A finite graph $G$ is $ϵ$-coherent if $\mathrm{\mid }G\mathrm{\mid }>1$, every vertex has degree less than $ϵ\mathrm{\mid }G\mathrm{\mid }$, and there are no disjoint anticomplete $A,B\subseteq V(G)$ with $\mathrm{\mid }A\mathrm{\mid },\mathrm{\mid }B\mathrm{\mid }\ge ϵ\mathrm{\mid }G\mathrm{\mid }$.

Let $\mathcal{B}=(B_1,\dots ,B_K)$ be a blockade. A minor of $\mathcal{B}$ is obtained by retaining some blocks in their original order and replacing each retained block by a nonempty subset. It is equicardinal when all its blocks have the same size. A copy of an ordered graph $J$ is $\mathcal{B}$-rainbow when its vertices lie in distinct blocks, in the prescribed order. Its support is the set of block indices it uses; the trace of $J$ is the family of all such supports.

For an integer $\tau \ge 1$, $\mathcal{B}$ is $\tau$-support-uniform if for every ordered tree $J$ of at most $\tau$ vertices its trace is either empty or contains every $\mathrm{\mid }J\mathrm{\mid }$-element set of block indices. For $0<\kappa \le 1$, it is $(\kappa ,\tau )$-support-invariant if every contraction of width at least $\kappa$ times its width has exactly the same trace for every such $J$.

Suppose the blocks have common size $W$. A set $X$ outside $B_i$ $\lambda$-covers $B_i$ if at least $\lambda W$ vertices of $B_i$ have a neighbor in $X$, and $\lambda$-misses $B_i$ if at least $\lambda W$ vertices of $B_i$ have none. The blockade is $\lambda$-concave if no $i<j<k$ and $X\subseteq V(\mathcal{B})\setminus (B_i\cup B_j\cup B_k)$ exist such that $X$ $\lambda$-covers $B_j$ and $\lambda$-misses both $B_i$ and $B_k$.

For integers $\delta \ge 2$, $\eta \ge 0$, let $T(\delta ,\eta )$ be the rooted complete $\delta$-ary tree of height $\eta$. A rainbow rooted tree is left-rainbow if its root is in its leftmost used block; right-rainbow is defined symmetrically.