= Solution
Spatial flatness constrains the total density, not matter alone. A flat model can have
$$
\boxed{\Omega_{m0}+\Omega_{\Lambda0}+\Omega_{r0}=1}
$$
with a substantial <dark energy> component. A positive <cosmological constant> has <pressure> $p_\Lambda=-\rho_\Lambda c^2$ and does not dilute like matter; it can produce the observed accelerated expansion.
If positive $\Lambda$ persists, $a(t)$ becomes exponential and the future <conformal time> interval
$$
\Delta\eta_{\rm future}=\int_{t_0}^{\infty}\frac{dt}{a(t)}
$$
is finite. For two comoving civilisations separated by $\chi$, each one-way signal consumes $\chi/c$. Thus a message completing $n$ legs at finite time requires
$$
\boxed{n\frac\chi c<\Delta\eta_{\rm future}.}
$$
Only finitely many legs, and hence finitely many full exchanges, can then occur. A reply to even the first new message requires $2\chi/c<\Delta\eta_{\rm future}$, the <cosmological round-trip communication horizon> condition. Depending on the density parameters, the initial outward message might arrive while its reply can never reach us, as in the right-hand panel of part <3/d>.
The source gives no matter and dark-energy fractions, so it does not specify an exact count for the $z=1$ pair in the real Universe. Flatness and lack of matter dominance alone do not prove a finite horizon: a radiation-dominated flat universe, for example, has divergent future <conformal time> if that power-law expansion persists. The finite-message conclusion uses persistent accelerated expansion such as positive constant $\Lambda$.
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