No four subsets forming an N

No four subsets forming an N

We survey results concerning the maximum size of a family F of subsets of an n-element set such that a certain configuration is avoided. When F avoids a chain of size two, this is just Sperner’s Theorem. Here we give bounds on how large F can be such that no four distinct sets A, B, C, D ∈ F satisfy A ⊂ B, C ⊂ B, C ⊂ D. In this case, the maximum size satisfies ( n b 2 c ) ( 1 + 1 n + Ω ( 1 n ))...

Universal cycles for k-subsets of an n-set

Generalized from the classic de Bruijn sequence, a universal cycle is a compact cyclic list of information. Existence of universal cycles has been established for a variety of families of combinatorial structures. These results, by encoding each object within a combinatorial family as a length-j word, employ a modified version of the de Bruijn graph to establish a correspondence between an Eule...

On Universal Cycles for k-Subsets of an n-Set

Collapsing with No Proper Extremal Subsets

This is a technical paper devoted to the investigation of collapsing of Alexandrov spaces with lower curvature bound. In a previous paper, the author defined a canonical stratification of an Alexandrov space by the so-called extremal subsets. It is likely that if the limit of a collapsing sequence has no proper extremal subsets, then the collapsing spaces are fiber bundles over the limit space....

The Structure of Maximum Subsets of {1, ..., n} with No Solutions to a+b = kc

If k is a positive integer, we say that a set A of positive integers is k-sum-free if there do not exist a, b, c in A such that a + b = kc. In particular we give a precise characterization of the structure of maximum sized k-sum-free sets in {1, . . . , n} for k ≥ 4 and n large.