  In this paper, we introduce the generic circular triangle-free graph $\mathbb
C_3$ and propose a finite axiomatization of its first order theory. In
particular, our main results show that a countable graph $G$ embeds into
$\mathbb C_3$ if and only if it is a $\{K_3, K_1 + 2K_2, K_1+C_5, C_6\}$-free
graph. As a byproduct of this result, we obtain a geometric characterization of
finite $\{K_3, K_1 + 2K_2, K_1+C_5, C_6\}$-free graphs, and the (finite) list
of minimal obstructions of unit Helly circular-arc graphs with independence
number strictly less than three.
  The circular chromatic number $\chi_c(G)$ is a refinement of the classical
chromatic number $\chi(G)$. We construct $\mathbb C_3$ so that a graph $G$ has
circular chromatic number strictly less than three if and only if $G$ maps
homomorphically to $\mathbb C_3$. We build on our main results to show that
$\chi_c(G) < 3$ if and only if $G$ can be extended to a $\{K_3, K_1 + 2K_2,
K_1+C_5, C_6\}$-free graph, and in turn, we use this result to reprove an old
characterization of $\chi_c(G) < 3$ due to Brandt (1999). Finally, we answer a
question recently asked by Guzm\'an-Pro, Hell, and Hern\'andez-Cruz by showing
that the problem of deciding for a given finite graph $G$ whether $\chi_c(G) <
3$ is NP-complete.
