In this document we study sets of NIP formulas in some theories of fields and valued fields, with a special focus on the sets of quantifier-free and existential formulas. First, we give a new proof of the fact that Separably Closed Valued Fields of any characteristic and any imperfection degree are NIP, and use this result to fill some gaps of a proof of the so-called NIP Transfer Theorem for henselian valued fields of equal characteristic. Second, we prove a variant of a theorem of Johnson: every positive characteristic valued field whose existential formulas are NIP is henselian, and generalize this result for finer sets of existential formulas, like positive existential with a given number of quantifiers, in type-definable fields. Finally, we set the ground for the finer question of transfer of NIP formulas of valued fields with bounded quantifier rank. Namely, we prove that for any henselian equicharacteristic valued field, any formula of quantifier rank at most $n\geq 1$ is NIP if and only if the same is true for the residue field and the value group, provided that the valued field is separably defectless Kaplansky and conditional on a multi-variable generalization of a well known statement about indiscernible sequences of singletons in ac-valued fields.
我们调查的界限拉姆齐定理的关系定义在NIP结构。将模型论方法应用于有限组合学，推广了Bukh和Matousek [B. Bukh，J. Matoušek.“Erdens-Szekeres-type statements：Ramsey function and decidability in dimension $1$“，Duke Mathematical Journal 163.12（2014）：2243-2270]从半代数情况到$\mathbb{R}$的任意多项式有界的$o$ -最小扩展，并且示出它在$\mathbb{R}_{\exp}$中不成立。这为$o$ -极小结构的多项式有界性提供了一个新的组合刻画.我们还证明了一个类似的关系定义在$P$ -最小的结构，特别是领域的$p$ -adics。推广[D. Conlon，J. Fox，J. Pach，B. Sudakov，A. Suk“Ramsey-type results for semi-algebraic relations”，Transactions of the American Mathematical Society 366.9（2014）：5043-5065]中，我们表明在远端结构中，$k$ -ary可定义关系的上界由高度$k-1$的指数塔给出。
