We prove a general lemma (inspired by of Holroyd and Talbot) about the connection largest cardinalities (or weight) structures satisfying some hereditary property substructures same property. use it to show how results concerning forbidden subposet problems in Boolean poset imply analogous subspaces finite vector space. also study generalized subspaces.

A Banach space has property (S) if every normalized weakly null sequence contains a subsequence equivalent to the unit vector basis of c0. We show that the equivalence constant can be chosen "uniformly", i.e., independent of the choice of the normalized weakly null sequence. Furthermore we show that a Banach space with property (S) has property (u). This solves in the negative the conjecture th...

An additive hereditary graph property is a class of simple graphs which is closed under unions, subgraphs and isomorphisms. If P1, . . . ,Pn are graph properties, then a (P1, . . . ,Pn)-decomposition of a graph G is a partition E1, . . . , En of E(G) such that G[Ei], the subgraph of G induced by Ei, is in Pi, for i = 1, . . . , n. The sum of the properties P1, . . . ,Pn is the property P1 ⊕ · ·...

We consider the following class of problems: Given a graph G, find a maximum vertex induced subgraph of G satisfying a nontrivial hereditary property π. We show that this problem cannot be approximated for any such property π, within a factor of n1−2 for any 2 > 0, unless NP = ZPP . This improves the result in [LY93] where it was shown that for any nontrivial hereditary property, the maximum su...

We consider natural graph property tests, which act entirely independently of the size of the graph being tested (not just having a number of queries independent of the size). We introduce the notion of graph properties being inflatable — closed under taking (balanced) blowups — and show that the query complexity of natural tests for a property are related to the degree to which it is approxima...

For a graph property X, let Xn be the number of graphs with vertex set {1, . . . , n} having property X, also known as the speed of X. A property X is called factorial if X is hereditary (i.e. closed under taking induced subgraphs) and n1 ≤ Xn ≤ n c2n for some positive constants c1 and c2. Hereditary properties with the speed slower than factorial are surprisingly well structured. The situation...

In this paper, we characterize both closed and strongly subobjects in the category of bounded uniform filter spaces introduce two notions closure operators which satisfy weakly hereditary, idempotent productive properties. We further each Tj (j= 0,1) using these examine that them form quotient-reflective subcategories spaces. Also, connected Finally, ultraconnected objects topological relations...