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Unformatted text preview: { PAGE } APPLICATIONS OF LARGE CARDINALS TO GRAPH THEORY by Harvey M. Friedman Department of Mathematics Ohio State University friedman@math.ohio-state.edu www.math.ohio-state.edu/~friedman August 22, 1997 revised October 23, 1997 TABLE OF CONTENTS Introduction 1. Decreasing classes of functions. 2. Jump free classes of functions. 3. Distance functions in graphs. 4. Williamsons inductive model. INTRODUCTION In [Fr97] we presented the first examples of statements in discrete and finite mathematics with a clear combinatorial meaning, which are proved using large cardinals, and shown to require them. The large cardinals in question are the subtle cardinals of finite order. Since then we have been engaged in the development of such results of greater relevance to mathematical practice. In January, 1997 we presented some new results of this kind involving what we call jump free classes of finite functions. This Jump Free Theorem is treated in section 2. Gill Williamson had the remarkable insight that the Jump Free Theorem could be applied to give information concerning various natural distance functions in subgraphs of a given graph. Williamson proposed several kinds of assertions in this vein, and proved that some of them do indeed follow from the Jump Free Theorem. We showed that some others also follow from the Jump Free Theorem, and also simplified and streamlined the applications. Williamson has informed us that these applications belong to a general class of problems of interest to a wide community of graph theorists, combinatorialists, and computer scientists. { PAGE } We also were able to prove that the more elementary of these applications of the Jump Free Theorem could be directly proved without the Jump Free Theorem using classical Ramsey theory. This established that these particular applications can be proved well within the usual axioms for mathematics. Next, Sam Buss and Williamson collaborated to extend our results to obtain more of the applications of the Jump Free Theorem just using classical Ramsey theory. Then we proved a general theorem on decreasing classes of functions, just using classical Ramsey theory, that covers this Buss/Williamson collaboration as special cases. It is still not yet clear whether the remaining, more sophisticated applications of the Jump Free Theorem can be proved within the usual axioms for mathematics. These more sophisticated applications can be stated with more and more structure of the kind considered standard in graph theory and computer science, and form a virtually open ended series of applications. They bear a clear technical resemblance to the Jump Free Theorem, and we consider a proof of the independence of at least some natural version of these applications to be within reach of our technology....
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applctogrph - { PAGE } APPLICATIONS OF LARGE CARDINALS TO...

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