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For $n > 2k geq 4$ we consider intersecting families $mathcal F$ consisting of $k$-subsets of ${1, 2, ldots, n}$. Let $mathcal I(mathcal F)$ denote the family of all distinct intersections $F cap F$, $F eq F$ and $F, Fin mathcal F$. Let $mathcal A$ consist of the $k$-sets $A$ satisfying $|A cap {1, 2, 3}| geq 2$. We prove that for $n geq 50 k^2$ $|mathcal I(mathcal F)|$ is maximized by $mathcal A$.
For a family $mathcal F$, let $mathcal D(mathcal F)$ stand for the family of all sets that can be expressed as $Fsetminus G$, where $F,Gin mathcal F$. A family $mathcal F$ is intersecting if any two sets from the family have non-empty intersection. I
A family $mathcal F$ has covering number $tau$ if the size of the smallest set intersecting all sets from $mathcal F$ is equal to $s$. Let $m(n,k,tau)$ stand for the size of the largest intersecting family $mathcal F$ of $k$-element subsets of ${1,ld
Let $r(k)$ denote the maximum number of edges in a $k$-uniform intersecting family with covering number $k$. ErdH{o}s and Lovasz proved that $ lfloor k! (e-1) rfloor leq r(k) leq k^k.$ Frankl, Ota, and Tokushige improved the lower bound to $r(k) geq
Wilfs Sixth Unsolved Problem asks for any interesting properties of the set of partitions of integers for which the (nonzero) multiplicities of the parts are all different. We refer to these as emph{Wilf partitions}. Using $f(n)$ to denote the number
In this paper, we give bounds on the dichromatic number $vec{chi}(Sigma)$ of a surface $Sigma$, which is the maximum dichromatic number of an oriented graph embeddable on $Sigma$. We determine the asymptotic behaviour of $vec{chi}(Sigma)$ by showing