17 research outputs found
On the Size of K-Cross-Free Families
Two subsets A,B of an n-element ground set X are said to be crossing, if none of the four sets A∩B, A\B, B\A and X\(A∪B) are empty. It was conjectured by Karzanov and Lomonosov forty years ago that if a family F of subsets of X does not contain k pairwise crossing elements, then |F|=O(kn). For k=2 and 3, the conjecture is true, but for larger values of k the best known upper bound, due to Lomonosov, is |F|=O(knlogn). In this paper, we improve this bound for large n by showing that |F|=Ok(nlog*n) holds, where log* denotes the iterated logarithm function. © 2018 János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Natur
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
Clones of pigmented words and realizations of special classes of monoids
Clones are generalizations of operads forming powerful instruments to
describe varieties of algebras wherein repeating variables are allowed in their
relations. They allow us in this way to realize and study a large range of
algebraic structures. A functorial construction from the category of monoids to
the category of clones is introduced. The obtained clones involve words on
positive integers where letters are pigmented by elements of a monoid. By
considering quotients of these structures, we construct a complete hierarchy of
clones involving some families of combinatorial objects. This provides clone
realizations of some known and some new special classes of monoids as among
others the variety of left-regular bands, bounded semilattices, and regular
band monoids.Comment: 41 page
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum