  We discuss the horizon problem in a universe dominated by fluid with negative
pressure. We show that for generally accepted value of nonrelativistic matter
energy density parameter $\Omega_{m0}<1$, the horizon problem can be solved
only if the fluid influencing negative pressure (the so-called ``X'' component)
violates the point-wise strong energy condition and if its energy density is
sufficiently large $(\Omega_{X 0}>1)$. The calculated value of the
$\Omega_{X0}$ parameter allowing for the solution of the horizon problem is
confronted with some recent observational data. Assuming that $p_X/\rho_X<-0.6$
we find that the required amount of the ``X'' component is not ruled out by the
supernova limits. Since the value of energy density parameter $\Omega_{v0}$ for
cosmological constant larger than 1 is excluded by gravitational lensing
observations the value of the ratio $p_X/\rho_X$ should lie between the values
-1 and -0.6 if the model has to be free of the horizon problem beeing at the
same time consistent with observations. The value of $\Omega_{X0}+\Omega_{m0}$
in the model is consistent with the constraints $0.2<\Omega_{\text{tot}}<1.5$
following from cosmic microwave background observations provided that
$\Omega_{m0}$ is low ($<0.2$).
