Аңдатпа
We find conditions such that every $$\lambda $$ -homogeneous model with small $$ \lambda $$ satisfying these conditions is homogeneous. As a corollary, we obtain conditions guaranteeing that the following implication holds: If $$T $$ is a theory, $$\mu >|T| $$ , and every model of $$T $$ of cardinality $$ \mu $$ is $$\omega _1$$ -homogeneous then every model of $$T$$ of sufficiently large cardinality is homogeneous.