102 research outputs found
A note on tilted Sperner families with patterns
Let and be two nonnegative integers with and . We call
a \textit{(p,q)-tilted Sperner family
with patterns on [n]} if there are no distinct with:
Long (\cite{L}) proved that the cardinality of a (1,2)-tilted Sperner family
with patterns on is
We improve and generalize this result, and prove that the cardinality of
every ()-tilted Sperner family with patterns on [] is Comment: 8 page
Generalized Tur\'an problems for disjoint copies of graphs
Given two graphs and , the maximum possible number of copies of in
an -free graph on vertices is denoted by . We investigate the
function , where denotes vertex disjoint copies of a fixed
graph . Our results include cases when is a complete graph, cycle or a
complete bipartite graph.Comment: 18 pages. There was a wrong statement in the first version, it is
corrected no
A discrete isodiametric result: the Erd\H{o}s-Ko-Rado theorem for multisets
There are many generalizations of the Erd\H{o}s-Ko-Rado theorem. We give new
results (and problems) concerning families of -intersecting -element
multisets of an -set and point out connections to coding theory and
classical geometry. We establish the conjecture that for such
a family can have at most members
Forbidden subposet problems for traces of set families
In this paper we introduce a problem that bridges forbidden subposet and
forbidden subconfiguration problems. The sets form a
copy of a poset , if there exists a bijection such that for any the relation implies
. A family of sets is \textit{-free} if
it does not contain any copy of . The trace of a family on a
set is .
We introduce the following notions: is
-trace -free if for any -subset , the family
is -free and is trace -free if it is
-trace -free for all . As the first instances of these problems
we determine the maximum size of trace -free families, where is the
butterfly poset on four elements with and determine the
asymptotics of the maximum size of -trace -free families for
. We also propose a generalization of the main conjecture of the area of
forbidden subposet problems
Rounds in a combinatorial search problem
We consider the following combinatorial search problem: we are given some
excellent elements of and we should find at least one, asking questions
of the following type: "Is there an excellent element in ?".
G.O.H. Katona proved sharp results for the number of questions needed to ask in
the adaptive, non-adaptive and two-round versions of this problem.
We verify a conjecture of Katona by proving that in the -round version we
need to ask queries for fixed and this is sharp.
We also prove bounds for the queries needed to ask if we want to find at
least excellent elements.Comment: 14 page
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