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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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Accepted
Non-trivial parity maps in graphs
I think $K_{n,n+1}$ works. Clearly each side must be monochromatic. So the side of odd size must be all $0$. But then the other side must be all $0$.