230 research outputs found
High dimensional Hoffman bound and applications in extremal combinatorics
One powerful method for upper-bounding the largest independent set in a graph
is the Hoffman bound, which gives an upper bound on the largest independent set
of a graph in terms of its eigenvalues. It is easily seen that the Hoffman
bound is sharp on the tensor power of a graph whenever it is sharp for the
original graph.
In this paper, we introduce the related problem of upper-bounding independent
sets in tensor powers of hypergraphs. We show that many of the prominent open
problems in extremal combinatorics, such as the Tur\'an problem for
(hyper-)graphs, can be encoded as special cases of this problem. We also give a
new generalization of the Hoffman bound for hypergraphs which is sharp for the
tensor power of a hypergraph whenever it is sharp for the original hypergraph.
As an application of our Hoffman bound, we make progress on the problem of
Frankl on families of sets without extended triangles from 1990. We show that
if then the extremal family is the star,
i.e. the family of all sets that contains a given element. This covers the
entire range in which the star is extremal. As another application, we provide
spectral proofs for Mantel's theorem on triangle-free graphs and for
Frankl-Tokushige theorem on -wise intersecting families
Essentially tight bounds for rainbow cycles in proper edge-colourings
An edge-coloured graph is said to be rainbow if no colour appears more than
once. Extremal problems involving rainbow objects have been a focus of much
research over the last decade as they capture the essence of a number of
interesting problems in a variety of areas. A particularly intensively studied
question due to Keevash, Mubayi, Sudakov and Verstra\"ete from 2007 asks for
the maximum possible average degree of a properly edge-coloured graph on
vertices without a rainbow cycle. Improving upon a series of earlier bounds,
Tomon proved an upper bound of for this question. Very
recently, Janzer-Sudakov and Kim-Lee-Liu-Tran independently removed the
term in Tomon's bound, showing a bound of . We prove an upper
bound of for this maximum possible average degree when
there is no rainbow cycle. Our result is tight up to the term, and so it
essentially resolves this question. In addition, we observe a connection
between this problem and several questions in additive number theory, allowing
us to extend existing results on these questions for abelian groups to the case
of non-abelian groups
On some interconnections between combinatorial optimization and extremal graph theory
The uniting feature of combinatorial optimization and extremal graph theory is that in both areas one should find extrema of a function defined in most cases on a finite set. While in combinatorial optimization the point is in developing efficient algorithms and heuristics for solving specified types of problems, the extremal graph theory deals with finding bounds for various graph invariants under some constraints and with constructing extremal graphs. We analyze by examples some interconnections and interactions of the two theories and propose some conclusions
Tur\'{a}n's inequality, nonnegative linearization and amenability properties for associated symmetric Pollaczek polynomials
An elegant and fruitful way to bring harmonic analysis into the theory of
orthogonal polynomials and special functions, or to associate certain Banach
algebras with orthogonal polynomials satisfying a specific but frequently
satisfied nonnegative linearization property, is the concept of a polynomial
hypergroup. Polynomial hypergroups (or the underlying polynomials,
respectively) are accompanied by -algebras and a rich, well-developed and
unified harmonic analysis. However, the individual behavior strongly depends on
the underlying polynomials. We study the associated symmetric Pollaczek
polynomials, which are a two-parameter generalization of the ultraspherical
polynomials. Considering the associated -algebras, we will provide
complete characterizations of weak amenability and point amenability by
specifying the corresponding parameter regions. In particular, we shall see
that there is a large parameter region for which none of these amenability
properties holds (which is very different to -algebras of locally compact
groups). Moreover, we will rule out right character amenability. The crucial
underlying nonnegative linearization property will be established, too, which
particularly establishes a conjecture of R. Lasser (1994). Furthermore, we
shall prove Tur\'{a}n's inequality for associated symmetric Pollaczek
polynomials. Our strategy relies on chain sequences, asymptotic behavior,
further Tur\'{a}n type inequalities and transformations into more convenient
orthogonal polynomial systems.Comment: Main changes towards first version: The part on associated symmetric
Pollaczek polynomials was extended (with more emphasis on Tur\'{a}n's
inequality and including a larger parameter region), and the part on little
-Legendre polynomials became a separate paper. We added several references
and corrected a few typos. Title, abstract and MSC class were change
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