Abstract. Let G be a finite graph with the non-k * -order property (essentially, a uniform finite bound on the size of an induced sub-half-graph). A major result of the paper applies model-theoretic arguments to obtain a stronger version of Szemerédi's regularity lemma for such graphs, in which there are no irregular pairs, the bounds are significantly improved, and each component satisfies an indivisibility condition:Theorem 5.18. Let k * ∈ N and therefore k * * (a constant depending on k * , but ≤ 2 k * +2 ) be given. Let G be a finite graph with the non-k * -order property. Then for any ǫ > 0 there exists m = m(ǫ) such that for all sufficiently large A ⊆ G, there is a partition Ai : i < i( * ) ≤ m of A into at most m pieces, where:(1) for all i, j < i( * ), ||Ai| − |Aj || ≤ 1 (2) all of the pairs (Ai, Aj) are (ǫ, ǫ)-uniform, meaning that for some truth value t = t(Ai, Aj) ∈ {0, 1}, for all but < ǫ|Ai| of the elements of |Ai|, for all but < ζ|Aj| of the elements of Aj , (aRb) ≡ t(Ai, Aj) (3) all of the pieces Ai are ǫ-excellent (an indivisibility condition, Definition 5.2 below ) (4) if ǫ < 1 2 k * * , then m ≤ (3 + ǫ) 8 ǫ k * * Motivation for this work comes from a coincidence of model-theoretic and graph-theoretic ideas. Namely, it was known that the "irregular pairs" in the statement of Szemerédi's regularity lemma cannot be eliminated, due to the counterexample of half-graphs. The results of this paper show in what sense this counterexample is the only essential difficulty. The proof is largely modeltheoretic (though written to be accessible to finite combinatorialists): arbitrarily large half-graphs coincide with model-theoretic instability, so in their absence, structure theorems and technology from stability theory apply. In addition to the theorem quoted, we give several other regularity lemmas with different advantages, in which the indivisibility condition on the components is improved (at the expense of letting the number of components grow with |G|) and extend some of these results to the larger class of graphs without the independence property.

We connect and solve two long-standing open problems in quite different areas: the model-theoretic question of whether S O P 2 SOP_2 is maximal in Keisler’s order, and the question from general topology/set theory of whether p = t \mathfrak {p} = \mathfrak {t} , the oldest problem on cardinal invariants of the continuum. We do so by showing these problems can be translated into instances of a more fundamental problem which we state and solve completely, using model-theoretic methods.

Abstract. We give the first (ZFC) dividing line in Keisler's order among the unstable theories, specifically among the simple unstable theories. That is, for any infinite cardinal λ for which there is µ < λ ≤ 2 µ , we construct a regular ultrafilter D on λ so that (i) for any model M of a stable theory or of the random graph,The non-saturation result relies on the notion of flexible ultrafilters. To prove the saturation result we develop a property of a class of simple theories, called Qr 1 , generalizing the fact that whenever B is a set of parameters in some sufficiently saturated model of the random graph, |B| = λ and µ < λ ≤ 2 µ , then there is a set A with |A| = µ so that any nonalgebraic p ∈ S(B) is finitely realized in A. In addition to giving information about simple unstable theories, our proof reframes the problem of saturation of ultrapowers in several key ways. We give a new characterization of good filters in terms of "excellence," a measure of the accuracy of the quotient Boolean algebra. We introduce and develop the notion of moral ultrafilters on Boolean algebras. We prove a so-called "separation of variables" result which shows how the problem of constructing ultrafilters to have a precise degree of saturation may be profitably separated into a more set-theoretic stage, building an excellent filter, followed by a more model-theoretic stage: building so-called moral ultrafilters on the quotient Boolean algebra, a process which highlights the complexity of certain patterns, arising from first-order formulas, in certain Boolean algebras.

Abstract. This paper contributes to the set-theoretic side of understanding Keisler's order. We consider properties of ultrafilters which affect saturation of unstable theories: the lower cofinality lcf(ℵ0, D) of ℵ0 modulo D, saturation of the minimum unstable theory (the random graph), flexibility, goodness, goodness for equality, and realization of symmetric cuts. We work in ZFC except when noted, as several constructions appeal to complete ultrafilters thus assume a measurable cardinal. The main results are as follows. First, we investigate the strength of flexibility, known to be detected by non-low theories. Assuming κ > ℵ0 is measurable, we construct a regular ultrafilter on λ ≥ 2 κ which is flexible but not good, and which moreover has large lcf(ℵ0) but does not even saturate models of the random graph. This implies (a) that flexibility alone cannot characterize saturation of any theory, however (b) by separating flexibility from goodness, we remove a main obstacle to proving non-low does not imply maximal. Since flexible is precisely OK, this also shows that (c) from a set-theoretic point of view, consistently, ok need not imply good, addressing a problem from Dow 1985. Second, under no additional assumptions, we prove that there is a loss of saturation in regular ultrapowers of unstable theories, and also give a new proof that there is a loss of saturation in ultrapowers of non-simple theories. More precisely, for D regular on κ and M a model of an unstable theory, M κ /D is not (2 κ ) + -saturated; and for M a model of a non-simple theory and λ = λ <λ , M λ /D is not λ ++ -saturated. In the third part of the paper, we investigate realization and omission of symmetric cuts, significant both because of the maximality of the strict order property in Keisler's order, and by recent work of the authors on SOP2. We prove that if D is a κ-complete ultrafilter on κ, any ultrapower of a sufficiently saturated model of linear order will have no (κ, κ)-cuts, and that if D is also normal, it will have a (κ + , κ + )-cut. We apply this to prove that for any n < ω, assuming the existence of n measurable cardinals below λ, there is a regular ultrafilter D on λ such that any D-ultrapower of a model of linear order will have n alternations of cuts, as defined below. Moreover, D will λ + -saturate all stable theories but will not (2 κ ) + -saturate any unstable theory, where κ is the smallest measurable cardinal used in the construction.