Abstract
The Probabilistic Method is a lasting legacy of the late Paul Erdős. We give two examples - both problems first formulated by Erdős in the 1960s with new results in the last few years and both with substantial open questions. Further in both examples we take a Computer Science vantagepoint, creating a probabilistic algorithm to create the object (coloring, packing respectively) and showing that with positive probability the created object has the desired properties.
Given m sets each of size n (with an arbitrary intersection pattern) we want to color the underlying vertices Red and Blue so that no set is monochromatic. Erdős showed this may always be done if m <2n-1, we give a recent argument of Srinivasan and Radhukrishnan that extends this to m <c2n√n/ ln n. One first colors randomly and then recolors the blemishes with a clever random sequential algorithm
In a universe of size N we have a family of sets, each of size k, such that each vertex is in D sets and any two vertices have only o(D) common sets. Asymptotics are for fixed k with N,D →∞. We want an asymptotic packing, a subfamily of N/k disjoint sets. Erdős and Hanani conjectured such a packing exists (in an important special case of asymptotic designs) and this conjecture was shown by Rödl. We give a simple proof of the speaker that analyzes the random greedy algorithm.
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© 2002 Springer-Verlag Berlin Heidelberg
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Spencer, J. (2002). Erdős Magic. In: Rajsbaum, S. (eds) LATIN 2002: Theoretical Informatics. LATIN 2002. Lecture Notes in Computer Science, vol 2286. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45995-2_3
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DOI: https://doi.org/10.1007/3-540-45995-2_3
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