Inflation / ERP’s / required return / earnings growth.

I understand required return  if R increases, you discount CF at higher rate. Lower price, therefore Lower P/E

Earnings growth increases, CF increase. price increase. P/E Increase
How does ERP and inflation impact the P/E growth? Let’s say both are increasing. (Is this intuition correct: higher inflation = higher prices = higher P/E)
And what about the ERP what is the intuition here.
Thanks
Maybe someone else can cite something straight out of the curriculum’s readings on this, but I’ll take a shot.
Increasing inflation (inflation in goods and services, not asset prices) = rising nominal interest rates, which increases discount rates, which reduces PV of cash flows, in turn reducing P/Es.
Increasing ERP = increasing discount rates (i.e.,think all the models used to est. required return on equity like CAPM, FFM, PSM, where req. return on equity is a positive function of ERP), which reduces PV of cash flows, in turn reducing P/Es.
Cheers Mamba. The ERP one makes sense when you put it in the CAPM function