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Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.

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Maximum number of colors for an optimal tiling which guarantees infinite paths

This question is a more applicable version of the question I've asked in mathexchange recently: What is the maximum number of colors we can use to color $N^2$ square sub-tiles of $N×N$ square block t …