Titlebook: Extremal Combinatorics; With Applications in Stasys Jukna Textbook 2011Latest edition Springer-Verlag Berlin Heidelberg 2011 Combinatorics.

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Photonics for the Knowledge AgeThe general framework for the . in combinatorics is the following: if we want to come up with an upper bound on the size of a set of objects, associate them with elements in a vector space . of relatively low dimension, and show that these elements are linearly independent; hence, we cannot have more objects in our set than the dimension of ..

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https://doi.org/10.1007/978-3-319-78524-0Linear independence is one of the most basic concepts in linear algebra. No less important are the concepts of orthogonality and rank. In this chapter we consider some combinatorial applications of these two concepts.

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CountingWe start with the oldest combinatorial tool — ..

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Advanced CountingWhen properly applied, the (double) counting argument can lead to more subtle results than those discussed in the previous chapter.

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Wallow

Systems of Distinct RepresentativesA . for a sequence of (not necessarily distinct) sets ..,..,…,.. is a sequence of distinct elements ..,..,…,.. such that ..∈.. for all .=1,2,…,..

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Blocking Sets and the DualityIn this chapter we will consider one of the most basic properties of set systems — their duality. The dual of a family . consists of all (minimal under set-inclusion) sets that intersect all members of .. Dual families play an important role in many applications, boolean function complexity being just one example.

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The Basic MethodThe general framework for the . in combinatorics is the following: if we want to come up with an upper bound on the size of a set of objects, associate them with elements in a vector space . of relatively low dimension, and show that these elements are linearly independent; hence, we cannot have more objects in our set than the dimension of ..

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