Unavoidable Multicoloured Families of Configurations
Richard P. Anstee, Linyuan Lu

TL;DR
This paper improves bounds on the size of set systems that necessarily contain certain multicoloured configurations, using Ramsey Theory to extend previous combinatorial results.
Contribution
It provides a tighter exponential bound on the size of set systems that guarantee the presence of specific multicoloured submatrix configurations, extending prior work by Balogh and Bollobás.
Findings
Improved the bound to $2^{ck^2}$ for the function $f(k)$.
Established the existence of specific $k imes k$ submatrices with coloured patterns.
Extended results to bounds on the number of columns avoiding certain repeated submatrices.
Abstract
Balogh and Bollob\'as [{\em Combinatorica 25, 2005}] prove that for any there is a constant such that any set system with at least sets reduces to a -star, an -costar or an -chain. They proved . Here we improve it to for some constant . This is a special case of the following result on the multi-coloured forbidden configurations at 2 colours. Let be given. Then there exists a constant so that a matrix with entries drawn from with at least different columns will have a submatrix that can have its rows and columns permuted so that in the resulting matrix will be either or (for some ), where is the matrix with 's on the diagonal and 's else where, the matrix with 's…
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Taxonomy
TopicsMachine Learning and Algorithms · Computability, Logic, AI Algorithms · Limits and Structures in Graph Theory
