Artin-Schreier extensions and combinatorial complexity in henselian valued fields

We give explicit formulas witnessing IP, IP \!n , 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 NIP \!n context one way of Anscombe–Jahnke’s classification of NIP henselian valued fields. As a corollary, we obtain that NIP \!n henselian valued fields with NIP residue field are NIP. We also discuss tameness results for NTP2 henselian valued fields.