I gave a talk at the University of Haifa on June 07, 2012, intended for general audience.
Title: Strong Colorings: the study of the failure of generalized Ramsey statements
Abstract: A strong coloring from $X$ to $Y$ is a function that transforms relatively thin subsets of $X$ into relatively fat subsets of $Y$. The first example of a strong coloring is due to Sierpinski (1933), who constructed a function from $\mathbb R^2$ into $\{0,1\}$ with the (anti-Ramsey) property that the image of any uncountable square $A^2$ equals $\{0,1\}$.
In the mid 1960’s, Erdos and his collaborators, utilized the Continuum Hypothesis to construct a function from $\mathbb R^2$ into $\mathbb R$ with the remarkable property that the image of any uncountable square $A^2$ equals $\mathbb R$. Ever since, the study of strong colorings has focused on constructing colorings for various sets without the aid of any additional set theoretic axioms.
In this talk, we shall survey the history of the theory of strong colorings, and report on our recent contributions to the theory.
This is joint work with Stevo Todorcevic.
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