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Branch of combinatorics with the philosophy that 'total disorder is impossible'. For example, Ramsey's theorem asserts that for each $n$, every sufficiently large graph either contains a clique of size $n$ or a stable set of size $n$.

2 votes
Accepted

Permutations of Grid Colorings

O(1) is impossible even if you drop the condition of no monochromatic rectangles, and even if you know that the two cells are always chosen within a given row. Suppose the length of the rows, $m$, …
Hugh Thomas's user avatar
  • 6,327
5 votes

Bound on cardinality of a union

A better solution than my previous one is max_{1\leq i \leq n} iN - {i \choose 2}N_2 (That is to say, we can simply consider only i of the sets instead of all n of them, and then apply my previous …
Hugh Thomas's user avatar
  • 6,327
2 votes

Bound on cardinality of a union

There is the obvious lower bound of nN - {n \choose 2}N_2. (I'm taking N_2 to be the bound on the size of an intersection of 2 sets; I'm not sure if that's what you meant.) I don't think it's poss …
Hugh Thomas's user avatar
  • 6,327