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We generalize some results concerning the classical notion of a spreading model to spreading models of order ξ. Among other results, we prove that the set $S{M}_{\xi }^w\left(X\right)$ of ξ-order spreading models of a Banach space X generated by subordinated weakly null ℱ-sequences endowed with the pre-partial order of domination is a semilattice. Moreover, if $S{M}_{\xi }^w\left(X\right)$ contains an increasing sequence of length ω then it contains an increasing sequence of length ω₁. Finally, if $S{M}_{\xi }^w\left(X\right)$ is uncountable, then it contains an antichain of size...

We prove a density version of the Carlson–Simpson Theorem. Specifically we show the following. For every integer $k\ge 2$ and every set $A$ of words over $k$ satisfying $\lim {\sup }_{n\to \infty }|A\cap {\left[k\right]}^n|/k^n>0$ there exist a word $c$ over $k$ and a sequence $\left(w_n\right)$ of left variable words over $k$ such that the set $c\cup \{c^{}{w}_0{\left(a_0\right)}^{}..{.}^{}{w}_n\left(a_n\right):n\in \mathbb{N}\text{and}{a}_0,...,a_n\in \left[k\right]\}$ 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.