47 research outputs found
Boolean algebras and Lubell functions
Let denote the power set of . A collection
\B\subset 2^{[n]} forms a -dimensional {\em Boolean algebra} if there
exist pairwise disjoint sets , all non-empty
with perhaps the exception of , so that \B={X_0\cup \bigcup_{i\in I}
X_i\colon I\subseteq [d]}. Let be the maximum cardinality of a family
\F\subset 2^X that does not contain a -dimensional Boolean algebra.
Gunderson, R\"odl, and Sidorenko proved that where .
In this paper, we use the Lubell function as a new measurement for large
families instead of cardinality. The Lubell value of a family of sets \F with
\F\subseteq \tsupn is defined by h_n(\F):=\sum_{F\in \F}1/{{n\choose |F|}}.
We prove the following Tur\'an type theorem. If \F\subseteq 2^{[n]} contains
no -dimensional Boolean algebra, then h_n(\F)\leq 2(n+1)^{1-2^{1-d}} for
sufficiently large . This results implies , where is an absolute constant independent of and . As a
consequence, we improve several Ramsey-type bounds on Boolean algebras. We also
prove a canonical Ramsey theorem for Boolean algebras.Comment: 10 page
Tree-width and dimension
Over the last 30 years, researchers have investigated connections between
dimension for posets and planarity for graphs. Here we extend this line of
research to the structural graph theory parameter tree-width by proving that
the dimension of a finite poset is bounded in terms of its height and the
tree-width of its cover graph.Comment: Updates on solutions of problems and on bibliograph
Tur\'an Numbers of Ordered Tight Hyperpaths
An ordered hypergraph is a hypergraph whose vertex set is linearly
ordered. We find the Tur\'an numbers for the -uniform -vertex tight path
(with vertices in the natural order) exactly when and
is even; our results imply
when
r\le s}(n,P^{(r)}_s)
remain open. For , we give a construction of an -uniform -vertex
hypergraph not containing which we conjecture to be asymptotically
extremal.Comment: 10 pages, 0 figure
Sublinear Longest Path Transversals and Gallai Families
We show that connected graphs admit sublinear longest path transversals. This
improves an earlier result of Rautenbach and Sereni and is related to the
fifty-year-old question of whether connected graphs admit constant-size longest
path transversals. The same technique allows us to show that -connected
graphs admit sublinear longest cycle transversals.
We also make progress toward a characterization of the graphs such that
every connected -free graph has a longest path transversal of size . In
particular, we show that the graphs on at most vertices satisfying this
property are exactly the linear forests.
Finally, we show that if the order of a connected graph is large relative
to its connectivity and , then each
vertex of maximum degree forms a longest path transversal of size
First-Fit is Linear on Posets Excluding Two Long Incomparable Chains
A poset is (r + s)-free if it does not contain two incomparable chains of
size r and s, respectively. We prove that when r and s are at least 2, the
First-Fit algorithm partitions every (r + s)-free poset P into at most
8(r-1)(s-1)w chains, where w is the width of P. This solves an open problem of
Bosek, Krawczyk, and Szczypka (SIAM J. Discrete Math., 23(4):1992--1999, 2010).Comment: v3: fixed some typo