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- W1780054780 abstract "Balogh and Bollobas [{em 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 $f(k) 0$. 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 $c_r$ so that a matrix with entries drawn from ${0,1,...,r-1}$ with at least $2^{c_rk^2}$ different columns will have a $ktimes k$ submatrix that can have its rows and columns permuted so that in the resulting matrix will be either $I_k(a,b)$ or $T_k(a,b)$ (for some $ane bin {0,1,..., r-1}$), where $I_k(a,b)$ is the $ktimes k$ matrix with $a$'s on the diagonal and $b$'s else where, $T_k(a,b)$ the $ktimes k$ 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 $t ktimes k$ matrix obtained by taking any one of the $k times k$ matrices above and repeating each column $t$ times. We use Ramsey Theory." @default.
- W1780054780 created "2016-06-24" @default.
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- W1780054780 date "2014-09-15" @default.
- W1780054780 modified "2023-09-27" @default.
- W1780054780 title "Unavoidable Multicoloured Families of Configurations" @default.
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