Extending the classical notion of the spreading model, the kspreading models of a Banach space are introduced, for every k ∈ N. The definition, which is based on the k-sequences and plegma families, reveals a new class of spreading sequences associated to a Banach space. Most of the results of the classical theory are stated and proved in the higher order setting. Moreover, new phenomena like the universality of the class of the 2-spreading models of c 0 and the composition property are established. As consequence, a problem concerning the structure of the k-iterated spreading models is solved. ∞ l=1 Plm l ([M ] k ). Notice that for l = 1 and every k ∈ N, we have Plm 1 ([M ] k ) = [M ] k . Moreover, for k = 1 and every l ∈ N, Plm l ([M ] 1 ) = [M ] l . In the sequel the elements of Plm 2 ([M ] k ) will be called plegma pairs in [M ] k .

Abstract. We introduce the higher order spreading models associated to a Banach space X. Their definition is based on F -sequences (xs) s∈F with F a regular thin family and the plegma families. We show that the higher order spreading models of a Banach space X form an increasing transfinite hierarchy (SM ξ (X)) ξ<ω 1 . Each SM ξ (X) contains all spreading models generated by F -sequences (xs) s∈F with order of F equal to ξ. We also provide a study of the fundamental properties of the hierarchy.

Given a hereditary family G of admissible graphs and a function λ(G) that linearly depends on the statistics of order-κ subgraphs in a graph G, we consider the extremal problem of determining λ(n, G), the maximum of λ(G) over all admissible graphs G of order n. We call the problem perfectly B-stable for a graph B if there is a constant C such that every admissible graph G of order n C can be made into a blow-up of B by changing at most C(λ(n, G)−λ(G)) n 2 adjacencies. As special cases, this property describes all almost extremal graphs of order n within o(n 2 ) edges and shows that every extremal graph of order n n 0 is a blow-up of B.We develop general methods for establishing stability-type results from flag algebra computations and apply them to concrete examples. In fact, one of our sufficient conditions for perfect stability is stated in a way that allows automatic verification by a computer. This gives a unifying way to obtain computer-assisted proofs of many new results.

We prove a density version of the Carlson-Simpson Theorem. Specifically we show the following. For every integer k 2 and every set A of words over k satisfying lim sup n→∞ |A ∩ [k] n | k n > 0 there exist a word c over k and a sequence (wn) of left variable words over k such that the set {c} ∪ c w 0 (a 0 ) ... wn(an) : n ∈ N and a 0 , ..., an ∈ [k]is contained in A.While the result is infinite-dimensional its proof is based on an appropriate finite and quantitative version, also obtained in the paper.2000 Mathematics Subject Classification: 05D10.

We give a purely combinatorial proof of the density Hales-Jewett Theorem that is modeled after Polymath's proof but is significantly simpler. In particular, we avoid the use of the equal-slices measure and work exclusively with the uniform measure.2000 Mathematics Subject Classification: 05D10.