Star and special-vertex obstructions force wonderfulness

Choose $a_0:=1$ in case 1. In case 2, let $a_0$ be the maximum order of a graph in $\{H\}\cup \bar{\mathcal{F}}$. By [L2], fix a constant $c\in (0,1)$ suitable for the corresponding obstruction hypothesis, and then choose $a\ge \max \{6,a_0,5\mathrm{/}c+1\}$.

Let $y\in (0,\frac{1}{2})$, let $G$ be a $\bar{\mathcal{F}}$-free graph, and let $\mathcal{B}=(B_1,\dots ,B_{\mathrm{\ell }})$ be an $(\mathrm{\ell },w)$-blockade satisfying the hypotheses from [L1] for the constant $a$. For each outside vertex $x\in V(G)\setminus V(\mathcal{B})$, define $I(x):=\{i\in [\mathrm{\ell }]:0<\mathrm{\mid }{N}_G(x)\cap B_i\mathrm{\mid }<\frac{1}{2}\mathrm{\mid }{B}_i\mathrm{\mid }\}$. Suppose first that $\mathrm{\mid }I(x)\mathrm{\mid }\le y\mathrm{\ell }$ for every such $x$. Then the number of pairs $(x,i)$ with $x\in V(G)\setminus V(\mathcal{B})$ and $i\in I(x)$ is at most $y\mathrm{\ell }\mathrm{\mid }V(G)\setminus V(\mathcal{B})\mathrm{\mid }\le y\mathrm{\ell }\mathrm{\mid }V(G)\mathrm{\mid }$. Averaging over the $\mathrm{\ell }$ indices, some $i\in [\mathrm{\ell }]$ is contained in at most $y\mathrm{\mid }V(G)\mathrm{\mid }$ of the sets $I(x)$. That is exactly the second conclusion from [L1].

It remains to consider the opposite case. Choose $v\in V(G)\setminus V(\mathcal{B})$ with $\mathrm{\mid }I(v)\mathrm{\mid }\ge y\mathrm{\ell }$. Let $\rho :[s]\to I(v)$ be the increasing bijection, where $s=\mathrm{\mid }I(v)\mathrm{\mid }$, put $C_j:=B_{\rho (j)}$, and form the auxiliary graph $J$ on $[s]$ by $ij\in E(J)$ if and only if $C_i$ is complete to $C_j$. The reordered family of blocks still has equal size, still satisfies $s\ge y\mathrm{\ell }$, and still satisfies the pairwise complete-or-mutually-$y^a$-sparse hypothesis. Therefore [L2] applies and gives a clique or stable set $R\subseteq [s]$ with $\mathrm{\mid }R\mathrm{\mid }\ge s^c$.

Put $R^{\mathrm{\prime }}:=\rho (R)\subseteq I(v)$. In the auxiliary graph on the original index set $I(v)$, the set $R^{\mathrm{\prime }}$ is a clique or stable set with $\mathrm{\mid }{R}^{\mathrm{\prime }}\mathrm{\mid }=\mathrm{\mid }R\mathrm{\mid }\ge \mathrm{\mid }I(v){\mathrm{\mid }}^c$. The original blockade $\mathcal{B}$ has length $\mathrm{\ell }\ge y^{-a}$, the subset $I(v)$ has size at least $y\mathrm{\ell }$, and $a\ge 5\mathrm{/}c+1$. Thus [L3] applies to $\mathcal{B}$, $I(v)$, and $R^{\mathrm{\prime }}$. It follows that ${\bigcup }_{i\in R^{\mathrm{\prime }}}{B}_i$ induces a $y^4$-restricted subgraph of $G$ whose size is at least the common block size, and therefore at least the width $w$ of $\mathcal{B}$. This is the first conclusion from [L1].

Step 1.2 gives the second wonderfulness outcome when no outside vertex belongs to many index sets $I(x)$, and step 3.1 gives the first outcome otherwise. Thus the constant $a$ from step 1.1 satisfies [L1], so $\mathcal{F}$ is wonderful.