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Old and New Results for the Weighted t-Intersection Problem via AK-Methods

  • Christian Bey
  • Konrad Engel
Chapter

Abstract

Let [n]: = {1, ..., n}, 2[n] be the power set of [n] and s ∈ [n]. A family F ⊆ 2[n] is called t-intersecting in [s] if
$$\left| {{X_1} \cap {X_2} \cap \left[ s \right]} \right| \geqslant t\,for\,all\,{X_1},{X_2}\, \in \,F.$$
Let ω: 2[n] → ℝ+ be a given weight function and
$${M_s}\left( {n,t;\omega } \right):\, = \max \left\{ {\omega \left( F \right)} \right.:F\,is\,t - \operatorname{int} er\sec ting\,in\left. {\,\left[ s \right]} \right\}.$$
For several weight functions, the numbers M n (n, t; ω) can be determined using three important methods of Ahlswede and Khachatrian: Generating Sets [2], Comparison Lemma [4], and Pushing—Pulling [3]. We survey these methods.
Also, sufficient conditions on ω for the equality
$${M_s}\left( {n,t;\omega } \right) = {M_n}\left( {n,t;\omega } \right)$$
are presented which simplify the method of Generating Sets. In addition, analogous conditions are given for the case that |∩ XF X| < t is required (nontrivial t-intersection).

Applications of these methods include new intersection theorems for chain- and star products.

Keywords

Weight Function Star Product Intersection Theorem Iterate Application Optimal Family 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Christian Bey
    • 1
  • Konrad Engel
    • 1
  1. 1.FB MathematikUniversität RostockRostockGermany

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