18,184 research outputs found
Intersection problems in combinatorics
With the publication of the famous Erdős-Ko-Rado Theorem in 1961, intersection problems became a popular area of combinatorics. A family of combinatorial objects is t-intersecting if any two of its elements mutually t-intersect, where the latter concept needs to be specified separately in each instance. This thesis is split into two parts; the first is concerned with intersecting injections while the second investigates intersecting posets.
We classify maximum 1-intersecting families of injections from {1, ..., k} to {1, ..., n}, a generalisation of the corresponding result on permutations from the early 2000s. Moreover, we obtain classifications in the general t>1 case for different parameter limits:
if n is large in terms of k and t, then the so-called fix-families, consisting of all injections which map some fixed set of t points to the same image points, are the only t-intersecting injection families of maximal size. By way of contrast, fixing the differences k-t and n-k while increasing k leads to optimal families which are equivalent to one of the so-called saturation families, consisting of all injections fixing at least r+t of the first 2r+t points, where r=|_ (k-t)/2 _|. Furthermore we demonstrate that, among injection families with t-intersecting and left-compressed fixed point sets, for some value of r the saturation family has maximal size .
The concept that two posets intersect if they share a comparison is new. We begin by classifying maximum intersecting families in several isomorphism classes of posets which are linear, or almost linear. Then we study the union of the almost linear classes, and derive a bound for an intersecting family by adapting Katona's elegant cycle method to posets. The thesis ends with an investigation of the intersection structure of poset classes whose elements are close to the antichain.
The overarching theme of this thesis is fixing versus saturation: we compare the sizes and structures of intersecting families obtained from these two distinct principles in the context of various classes of combinatorial objects
On the union of intersecting families
A family of sets is said to be \emph{intersecting} if any two sets in the
family have nonempty intersection. In 1973, Erd\H{o}s raised the problem of
determining the maximum possible size of a union of different intersecting
families of -element subsets of an -element set, for each triple of
integers . We make progress on this problem, proving that for any
fixed integer and for any , if is
an -element set, and , where each is an intersecting
family of -element subsets of , then , with equality only if $\mathcal{F} = \{S \subset X:\ |S|=k,\
S \cap R \neq \emptyset\}R \subset X|R|=ro(1)k < (3-\sqrt{5})n/2r=2$. Our proof utilises an
isoperimetric, influence-based method recently developed by Keller and the
authors.Comment: 13 pages. Updated references, expositional changes and minor
corrections following the helpful comments of an anonymous refere
On the number of maximal intersecting k-uniform families and further applications of Tuza's set pair method
We study the function which denotes the number of maximal
-uniform intersecting families . Improving a
bound of Balogh at al. on , we determine the order of magnitude of
by proving that for any fixed , holds. Our proof is based on Tuza's set pair
approach.
The main idea is to bound the size of the largest possible point set of a
cross-intersecting system. We also introduce and investigate some related
functions and parameters.Comment: 11 page
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