Suppose I perform an RNA-SEQ experiment, and I perform three different comparisons in it using, say, DESeq2. (i.e. WT timepoint1 vs MUT timepoint1; WT timepoint2 vs WT timepoint1; MUT timepoint2 vs MUT timepoint1).
In each comparison, there are many genes on which a statistical test is being performed; so within each comparison, a correction for multiple hypotheses has to be applied.
But I perform three comparisons, not one. The chance of falsely rejecting H0 for genes increases with each comparison that is being performed. So should I correct for multiple hypothesis due to the number of comparisons, and why not, if not?
To further play the devil's advocate : suppose that not only I perform this experiment, but another laboratory tries to perform exactly the same experiment. That laboratory also performs a statistical test, attempting to replicate my experiment. Every time we perform a statistical test, we increase the chance of a false positive. So why would the replication of the experiment by another lab not count as a reason to correct for multiple hypotheses?
In other words, what are the necessary and sufficient conditions for applying a multiple hypothesis correction?