  We establish a novel connection between the well-known chromatic threshold
problem in extremal combinatorics and the celebrated $(p,q)$-theorem in
discrete geometry. In particular, for a graph $G$ with bounded clique number
and a natural density condition, we prove a $(p,q)$-theorem for an abstract
convexity space associated with $G$. Our result strengthens those of Thomassen
and Nikiforov on the chromatic threshold of cliques. Our $(p,q)$-theorem can
also be viewed as a $\chi$-boundedness result for (what we call) ultra maximal
$K_r$-free graphs.
  We further show that the graphs under study are blow-ups of constant size
graphs, improving a result of Oberkampf and Schacht on homomorphism threshold
of cliques. Our result unravels the cause underpinning such a blow-up
phenomenon, differentiating the chromatic and homomorphism threshold problems
for cliques. It implies that for the homomorphism threshold problem, rather
than the minimum degree condition usually considered in the literature, the
decisive factor is a clique density condition on co-neighborhoods of vertices.
More precisely, we show that if an $n$-vertex $K_{r}$-free graph $G$ satisfies
that the common neighborhood of every pair of non-adjacent vertices induces a
subgraph with $K_{r-2}$-density at least $\varepsilon>0$, then $G$ must be a
blow-up of some $K_r$-free graph $F$ on at most
$2^{O(\frac{r}{\varepsilon}\log\frac{1}{\varepsilon})}$ vertices. Furthermore,
this single exponential bound is optimal. We construct examples with no
$K_r$-free homomorphic image of size smaller than
$2^{\Omega_r(\frac{1}{\varepsilon})}$.
