Fixed points on null and tame flows for groups of automorphisms

Using a generalization of the Kechris-Pestov-Todorčević correspondence due to Nguyen Van Thé we obtain fixed point theorems for null and tame actions of groups of the form $\mathrm{A}\mathrm{u}\mathrm{t}(\mathcal{F})$, where $\mathcal{F}$ is a Fraïssé structure. In particular we show that if $\mathrm{A}\mathrm{g}\mathrm{e}(\mathcal{F})$ is a free joint embedding class, then every null flow $\mathrm{A}\mathrm{u}\mathrm{t}(\mathcal{F})\curvearrowright X$ has a fixed point, while if $\mathrm{A}\mathrm{g}\mathrm{e}(\mathcal{F})$ is a free amalgamation class, then every tame flow $\mathrm{A}\mathrm{u}\mathrm{t}(\mathcal{F})\curvearrowright X$ has a fixed point.