What's the difference between rectangles and squares?

Ramsey's theorem states that for every symmetric partition of the square X² of an infinite set X into finitely many cells, there would be a cell that covers the square Y² of some infinite subset Y of X. Erdos and his collaborators considered various generalization of Ramsey's statement (replacing "infinite" with "uncountable", "finitely many" with "countably many", "square" with "rectangle"), and demonstraed the (consistent) failure of many of them. In this talk, we shall present this line of research, with the goal of elucidating the difference between rectangles and squares.

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