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Published byDarren Chase Modified over 9 years ago
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Constructing Ramsey Graphs Gil Cohen (or Two-source dispersers for polylog-entropy and improved Ramsey graphs)
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Ramsey graphs Ramsey theory concerns the unavoidable presence of local structure in globally unstructured objects. Ramsey’s Theorem (1928) Definition Theorem (Erdős 1947)
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Previous work Construction [Erdos’47] (non constructive) [Abbott’72] [Nagy’75] [Frankl’77] [Chung’81] [FranklWilson’81, Alon’98, Grolmusz’01, Barak’06]
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Previous work Bipartite construction Folklore [PudlakRodl’04] [BarakKindlerShaltielSudakovWigderson’05] [BarakRaoShaltielWigderson’06]
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Our contribution Main result Soon afterwards matched by [ChattopadhyayZuckerman’15] using independent techniques.
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Roadmap * The problem at hand * Rest of the talk – proof strategy in high-level * Switching lingo * Let’s continue from there GraphsFunctions SetsRandom variables
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Dispersers Definition (ChorGoldreich 1985) Definition Lemma Main result in dispersers’ lingo
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Block-sources Definition (ChorGoldreich 1985) Theorem (Li 2015)
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Subsources and a chain-like rule Definition Unfortunately, there is no chain rule for min-entropy. Nevertheless, some sort of a chain-like rule can be salvaged! Chain-like rule for min-entropy
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Roadmap * Block-sources and Li’s theorem * A chain-like rule for min-entropy * The problem at hand * Rest of the talk – proof strategy in high-level * Switching lingo * The high-level strategy
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The high-level strategy
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Roadmap * The high-level strategy * Block-sources and Li’s theorem * A chain-like rule for min-entropy * The problem at hand * Rest of the talk – proof strategy in high-level * Switching lingo * Entropy trees
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Entropy trees Claim Corollary
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Entropy trees Claim
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Entropy trees
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F
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F
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F F
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F F B Claim
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Roadmap * Revisiting the high-level strategy * The high-level strategy * Block-sources and Li’s theorem * A chain-like rule for min-entropy * The problem at hand * Rest of the talk – proof strategy in high-level * Switching lingo * Entropy trees
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The high-level strategy – revisited The high-level strategy - revisited Unfortunately, we don’t know how to implement the second step. We overcome this by enforcing even more structure on the sources.
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The structure we enforce
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A more detailed strategy
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F
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Roadmap * Revisiting the high-level strategy * The high-level strategy * Block-sources and Li’s theorem * A chain-like rule for min-entropy * The problem at hand * Rest of the talk – proof strategy in high-level * Switching lingo * Entropy trees * Quite a lot had to be omitted… * Identifying the entropy paths
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Summary and open problems Erdős’ challenge Our result Next natural goal * Quantitative improvement * Even a weaker notion of explicitness is interesting. * A simple construction (like [FranklWilson’81] ), even with weaker parameters. Thanks!
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