$\mathscr C\overset{u.c.m}{\longrightarrow}\mathscr P$ iff $\exists S\subset_{fin} C$ on which
no non-trivial identity holds.
Suppose that $\mathscr C\overset{m}{\longrightarrow}\mathscr P$. Does $\mathscr C\overset{\color{red}{u.c.}m}{\longrightarrow}\mathscr P$?
Suppose that $\mathscr C$ satisfies non-trivial height 1 identities locally.
Does it globally satisfy a non-trivial set of height 1 identities?
Motivated by study of constraint satisfaction problems over $\omega$-categorical structures.
$\Csp(\Bbb G)$: input finite $\Bbb H$, does there exist a homomorphism $\Bbb H\to\Bbb G$?
If $\Bbb G$ is $\omega$-categorical, complexity of $\Csp(\Bbb G)\leftrightarrow$ polymorphisms of $\Bbb G$
Conjecture: If $\Pol(\Bbb G)\mathrel{\rlap{\hskip .5em/}}\overset{u.c.m}{\longrightarrow}\mathscr P$, then $\Csp(\Bbb G)$ is in P. [Barto-Opršal-Pinsker, '15]
Known: If $\Pol(\Bbb G)\overset{u.c.m}{\longrightarrow}\mathscr P$, then $\Csp(\Bbb G)$ is NP-hard.
If $\Pol(\Bbb G)\longrightarrow\mathscr P$, then there exist $c_1,\dots,c_n$ such that
$\Pol(\Bbb G,c_1,\dots,c_n)\overset{u.c.}\longrightarrow\mathscr P$.
Topology is irrelevant [Barto-Pinsker]
Previously...
Jakub's talk
Graph $\Bbb H \rightarrow$ identities $\Sigma_{\Bbb H}$ of height 1.
If $\Bbb H$ is not 3-colorable then $\Sigma_{\Bbb H}$ is non-trivial.
A sequence $\Bbb H_1,\Bbb H_2,\dots$ such that $\Sigma_{\Bbb H_n}\Rightarrow \Sigma_{\Bbb H_{n+1}}$ and such that any non-trivial $\Sigma$ implies $\Sigma_{\Bbb H_n}$ for some $n$.
Manuel's talk
For all $n$, $\Sigma_{\Bbb H_{n+1}}\not\Rightarrow\Sigma_{\Bbb H_n}$.
There exists a graph $\Bbb G_n = (V, E_n)$ such that $\Pol(\Bbb G_n)$ does not satisfy $\Sigma_{\Bbb H_n}$
but $\Pol(\Bbb G_n)\mathrel{\rlap{\hskip .5em/}}\overset{m}{\longrightarrow}\mathscr P$.
$\Bbb G_n$ is $\omega$-categorical and has a finitely bounded homogeneous expansion by finitely many relations.
Goal: obtain an $\omega$-categorical graph $\Bbb G$ such that
$\Pol(\Bbb G)\mathrel{\rlap{\hskip .5em/}}\overset{u.c.m}{\longrightarrow}\mathscr P$ and $\Pol(\Bbb G)\overset{m}{\longrightarrow}\mathscr P$. Idea: superpose the $\Bbb G_n$, obtain something of the form $(V; E_1, E_2, \dots)$.
$f_{\Bbb G}(n)=$ number of orbits of $n$-tuples under $\Aut(\Bbb G)$
By sparsifying, one gets $f_{\Bbb G}(n)<2^{2^n}$
$\Pol(\Bbb G)$ satisfies
\[\alpha s(x,y,x,z,y,z)\approx \beta s(y,x,z,x,z,y)\]
so that $\Pol(\Bbb G)\mathrel{\rlap{\hskip .5em/}}\longrightarrow\mathscr P$.
If $\Bbb G$ has less than double-exponential orbit-growth
and $\Pol(\Bbb G)\mathrel{\rlap{\hskip .5em/}}\longrightarrow\mathscr P$,
then $\Pol(\Bbb G)\mathrel{\rlap{\hskip .5em/}}\overset{u.c.m}{\longrightarrow}\mathscr P$.