Patches is correct. There were 1,000,000 soldiers in Europe in 1918, and more than double that served there, many after the armistice.
It rapidly reduced to approx 100,000 in the early 20s, up to approxc 180,000 in 1938-39. Most divisions operated at well less than required TO & Es. Most cavalry "divisions" were really no more than the size of polo teams, with a HQ staff only.
Can't promote if there are no slots; you could only hope for someone to leave or die.