Unavoidable Multicoloured Families of Configurations
نویسندگان
چکیده
Balogh and Bollobás [Combinatorica 25, 2005] prove that for any k there is a constant f(k) such that any set system with at least f(k) sets reduces to a k-star, an k-costar or an k-chain. They proved f(k) < (2k)2 k . Here we improve it to f(k) < 2ck 2 for some constant c > 0. This is a special case of the following result on the multi-coloured forbidden configurations at 2 colours. Let r be given. Then there exists a constant cr so that a matrix with entries drawn from {0, 1, . . . , r − 1} with at least 2crk2 different columns will have a k× k submatrix that can have its rows and columns permuted so that in the resulting matrix will be either Ik(a, b) or Tk(a, b) (for some a 6= b ∈ {0, 1, . . . , r − 1}), where Ik(a, b) is the k × k matrix with a’s on the diagonal and b’s else where, Tk(a, b) the k× k matrix with a’s below the diagonal and b’s elsewhere. We also extend to considering the bound on the number of distinct columns, given that the number of rows is m, when avoiding a tk × k matrix obtained by taking any one of the k×k matrices above and repeating each column t times. We use Ramsey Theory. ∗Research supported in part by NSERC, some work done while visiting the second author at USC. †Research supported in part by NSF grant DMS 1300547 and ONR grant N00014-13-1-0717.
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Multicoloured extremal problems
Many problems in extremal set theory can be formulated as finding the largest set system (or r-uniform set system) on a fixed ground set X that does not contain some forbidden configuration of sets. We shall consider multicoloured versions of such problems, defined as follows. Given a list of set systems, which we think of as colours, we call another set system multicoloured if for each of its ...
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