A note on stability and NIP in one variable

A theory is NIP (resp. stable) if and only if every formula with parameters in two single variables is NIP (resp. does not have the order property). It is well known that a theory is NIP (resp. stable) if and only if all formulas φ(x; ȳ), where x is a singleton, are NIP (resp. stable). It is also well known that one cannot ask for both x and y to be single variables, as one sees by considering a random ternary hypergraph for instance. In this short note, we prove that we can in fact take both x and y to be single variables at the cost of allowing φ to have parameters. Theorem. Let T be a first order theory. Assume that for every model M |= T , for every formula φ(x; y) ∈ L(M) is NIP (resp. does not have the order property), where x and y are single variables and φ has parameters from M , then T is NIP (resp. stable). We first prove the stable case. Assume that T is unstable and let φ(x̄; ȳ) be a formula with the order property. If |x̄| = |ȳ| = 1, we are done. Otherwise by symmetry, we can assume |ȳ| > 1. We will construct a formula ψ(x̄; ȳ) with the order property, where |ȳ| < |ȳ| and having parameters in some model of T . We can then conclude by induction. For the induction to go through, we need to allow for φ to have parameters. However, we can name those parameters by adding constants to the language and hence without loss of generality, we assume that φ is without parameters. By Ramsey and compactness, we can find in some modelM of T an indiscernible sequence (āi, b̄i : i ∈ Q) such that |= φ(āi; b̄j) ⇐⇒ i < j. Write Q = Q \ {0} and let b̄0 = b 0 0ˆ̄b ′ 0, where b 0 0 is a singleton. Assume first that the sequence (āi : i ∈ Q ) is indiscernible over b00. Then for any k ∈ Q , there is b̄′k such that, for i ∈ Q \ {k}, we have |= φ(āi; b 0 0, b̄ ′ k) ⇐⇒ i < k. Thus the formula ψ(x̄; ȳ) := φ(x̄; b00, ȳ ) has the order property and we are done. Assume now that the sequence (āi : i ∈ Q ) is not indiscernible over b00. By construction, we know that the two sequences (āi : i < 0) and (āi : i > 0) are mutually indiscernible over b00. For a finite set A ⊂ Q , let QA ⊆ Q ∗ be the set of points which are either positive less than all positive elements in A or negative, greater than all negative elements in A. Let also āA denote the union of the tuples āi, i ∈ A. This research was partly supported by NSF (grants no. 1665491 and 1848562).