1. Constructing Ramsey Graphs Gil Cohen (or Two-source dispersers for polylog-entropy and improved Ramsey graphs)

2. Ramsey graphs Ramsey theory concerns the unavoidable presence of local structure in globally unstructured objects. Ramsey’s Theorem (1928) Definition Theorem (Erdős 1947)

3. Previous work Construction [Erdos’47] (non constructive) [Abbott’72] [Nagy’75] [Frankl’77] [Chung’81] [FranklWilson’81, Alon’98, Grolmusz’01, Barak’06]

4. Previous work Bipartite construction Folklore [PudlakRodl’04] [BarakKindlerShaltielSudakovWigderson’05] [BarakRaoShaltielWigderson’06]

5. Our contribution Main result Soon afterwards matched by [ChattopadhyayZuckerman’15] using independent techniques.

6. Roadmap * The problem at hand * Rest of the talk – proof strategy in high-level * Switching lingo * Let’s continue from there GraphsFunctions SetsRandom variables

7. Dispersers Definition (ChorGoldreich 1985) Definition Lemma Main result in dispersers’ lingo

8. Block-sources Definition (ChorGoldreich 1985) Theorem (Li 2015)

9. 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

10. 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

11. The high-level strategy

14. 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

15. Entropy trees Claim Corollary

16. Entropy trees Claim

17. Entropy trees

18. F

19. F

20. F F

21. F F B Claim

22. 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

23. 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.

24. The structure we enforce

27. A more detailed strategy

28. F

29. 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

30. 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!