I have a question involving P-Values in a specific context. Suppose you have an 1-Way ANOVA with k groups. If you obtain a significant p-value (<0.05) for the omnibus test, you usually query further with multiple comparisons to find exactly which of the k groups differ from each other. Is there a mathematical formulation of this? In general, if you have a significant omnibus, should you have at least one significant comparison? What's the theory/mathematical result behind this?

Of course I know in practice this always isn't the case (a significant omnibus but no significant multiple comparison)

Just thinking, please post if you have some insights!