Artin-Schreier extensions and combinatorial complexity in henselian valued fields

We give explicit formulas witnessing IP, \IPn or TP2 in fields with Artin-Schreier extensions. We use them to control $p$-extensions of mixed characteristic henselian valued fields, allowing us most notably to generalize to the \NIPn context one way of Anscombe-Jahnke's classification of NIP henselian valued fields. As a corollary, we obtain that \NIPn henselian valued fields with NIP residue field are NIP. We also discuss tameness results for NTP2 henselian valued fields.