Disjointly representing set systems

AbstractA family F of sets is s-disjointly representable if there is a family S of disjoint sets each of size s such that every F∈F contains some S∈S. Let f(r,s) be the minimum size of a family F of r-sets which is not s-disjointly representable. We give upper and lower bounds on f(r,s) which are within a constant factor when s is fixed

Banach spaces without minimal subspaces

We prove three new dichotomies for Banach spaces \`a la W.T. Gowers' dichotomies. The three dichotomies characterise respectively the spaces having no minimal subspaces, having no subsequentially minimal basic sequences, and having no subspaces crudely finitely representable in all of their subspaces. We subsequently use these results to make progress on Gowers' program of classifying Banach spaces by finding characteristic spaces present in every space. Also, the results are used to embed any partial order of size $\aleph_1$ into the subspaces of any space without a minimal subspace ordered by isomorphic embeddability

Set systems related to a house allocation problem

We are given a set of buyers, a set of houses, and for each buyer a preference list, i.e., an ordering of the houses. A house allocation is an injective mapping from to , and is strictly better than another house allocation if for every buyer , does not come before in the preference list of . A house allocation is Pareto optimal if there is no strictly better house allocation. Let be the image of i.e., the set of houses sold in the house allocation . We are interested in the largest possible cardinality of the family of sets for Pareto optimal mappings taken over all sets of preference lists of buyers and all sets of houses. This maximum exists since in a Pareto optimal mapping with buyers, each buyer will always be assigned one of their top choices. We improve the earlier upper bound on given by Asinowski et al. (2016), by making a connection between this problem and some problems in extremal set theory

Modified mixed Tsirelson spaces

We study the modified and boundedly modified mixed Tsirelson spaces $T_M[({\cal F}_{k_n},\theta_n)_{n=1}^{\infty }]$ and $T_{M(s)}[({\cal F}_{k_n},\theta_n)_{n=1}^{\infty }]$ respectively, defined by a subsequence $({\cal F}_{k_n})$ of the sequence of Schreier families $({\cal F}_n)$. These are reflexive asymptotic $\ell_1$ spaces with an unconditio- nal basis $(e_i)_i$ having the property that every sequence $\{ x_i\}_{i=1}^n$ of normalized disjointly supported vectors contained in $\langle e_i\rangle_{i=n}^{\infty }$ is equivalent to the basis of $\ell_1^n$. We show that if $\lim\theta_n^{1/n}=1$ then the space $T[({\cal F}_n,\theta_n) _{n=1}^{\infty }]$ and its modified variations are totally incomparable by proving that $c_0$ is finitely disjointly representable in every block subspace of $T[({\cal F}_n, \theta_n)_{n=1}^{\infty }]$. Next, we present an example of a boundedly modified mixed Tsirelson space $X_{M(1),u}$ which is arbitrarily distortable. Finally, we construct a variation of the space $X_{M(1),u}$ which is hereditarily indecomposable

Examples of k-iterated spreading models

It is shown that for every $k\in\mathbb{N}$ and every spreading sequence $\{e_n\}_{n\in\mathbb{N}}$ that generates a uniformly convex Banach space $E$, there exists a uniformly convex Banach space $X_{k+1}$ admitting $\{e_n\}_{n\in\mathbb{N}}$ as a $k+1$-iterated spreading model, but not as a $k$-iterated one.Comment: 16 pages, no figure

On the number of permutatively inequivalent basic sequences in a Banach space

AbstractLet X be a Banach space with a Schauder basis (en)n∈N. The relation E0 is Borel reducible to permutative equivalence between normalized block-sequences of (en)n∈N or X is c0 or ℓp saturated for some 1⩽p<+∞. If (en)n∈N is shrinking unconditional then either it is equivalent to the canonical basis of c0 or ℓp, 1<p<+∞, or the relation E0 is Borel reducible to permutative equivalence between sequences of normalized disjoint blocks of X or of X∗. If (en)n∈N is unconditional, then either X is isomorphic to ℓ2, or X contains 2ω subspaces or 2ω quotients which are spanned by pairwise permutatively inequivalent normalized unconditional bases