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Sunflowers in Set Systems of Bounded Dimension

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Abstract

Given a family \({\mathcal {F}}\) of k-element sets, \(S_1,\ldots ,S_r\in {\mathcal {F}}\) form an r-sunflower if \(S_i \cap S_j =S_{i'} \cap S_{j'}\) for all \(i \ne j\) and \(i' \ne j'\). According to a famous conjecture of Erdős and Rado (JAMA 35: 85–90, 1960), there is a constant \(c=c(r)\) such that if \(|{\mathcal {F}}|\ge c^k\), then \({\mathcal {F}}\) contains an r-sunflower. We come close to proving this conjecture for families of bounded Vapnik-Chervonenkis dimension, \(\text {VC-dim}({\mathcal {F}})\le d\). In this case, we show that r-sunflowers exist under the slightly stronger assumption \(|{\mathcal {F}}|\ge 2^{10k(dr)^{2\log ^{*} k}}\). Here, \(\log ^*\) denotes the iterated logarithm function. We also verify the Erdős-Rado conjecture for families \({\mathcal {F}}\) of bounded Littlestone dimension and for some geometrically defined set systems.

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References

  1. Abbott, H.L., Hanson, D., Sauer, N.: Intersection theorems for systems of sets. J. Combin. Theory Ser. A 12, 381–389 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ajtai, M., Chvátal, V., Newborn, M., Szemerédi, E.: Crossing-free subgraphs, in: Theory and Practice of Combinatorics 60, Ann. Discrete Math., North-Holland Math. Stud. 12, North-Holland, Amsterdam, pp. 9–12 (1982)

  3. Alon, N., Holzman, R.: Near-sunflowers and focal families. Israel J. Math. (2020). https://doi.org/10.48550/arXiv.2010.05992

  4. Alon, N., Livni, R., Malliaris, M., Moran, S.: Private PAC learning implies finite Littlestone dimension, Proc. STOC, pp. 852–860 (2019)

  5. Alweiss, R., Lovett, S., Wu, K., Zhang, J.: Improved bounds for the sunflower lemma. Ann. Math. 194, 795–815 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  6. Axenovich, M., Fon-Der-Flaass, D., Kostochka, A.: On set systems without weak 3-\(\Delta \)-subsystems. Discrete Math. 138, 57–62 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ben-David, S., Pál, D., Shalev-Shwartz, S.: Agnostic online learning, in COLT, (2009)

  8. Buzaglo, S., Holzman, R., Pinchasi, R.: On \(s\)-intersecting curves and related problems, Symposium on Computational Geometry, pp. 79–84 (2008)

  9. Chase, H., Freitag, J.: Model theory and machine learning. Bull. Symb. Log. 25, 319–332 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  10. Clarkson, K.L.: Applications of random sampling in computational geometry, II, Symposium on Computational Geometry, pp. 1–11 (1988)

  11. Deza, M., Frankl, P.: Every large set of equidistant \((0,+1,-1)\)-vectors forms a sunflower. Combinatorica 1, 225–231 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ding, G., Seymour, P., Winkler, P.: Bounding the vertex cover number of a hypergraph. Combinatorica 14, 23–34 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  13. Erdős, P., Rado, R.: Intersection theorems for systems of sets. J. Lond. Math. Soc. 35, 85–90 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  14. Erdős, P., Szemerédi, E.: Combinatorial properties of systems of sets. J. Combin. Theory Ser. A 24, 308–313 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fox, J., Pach, J., Suk, A.: Erdős-Hajnal conjecture for graphs with bounded VC-dimension. Discrete Comput. Geom. 61, 809–829 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fox, J., Pach, J., Suk, A.: Bounded VC-dimension implies the Schur-Erdős conjecture, Symposium on Computational Geometry, pp. 46:1–46:8 (2020)

  17. Frankl, P.: On the trace of finite sets. J. Combin. Theory Ser. A 34, 41–45 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kostochka, A.V.: An intersection theorem for systems of sets. Random Struct. Algorithms 9, 213–221 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  19. Littlestone, N.: Learning quickly when irrelevant attributes abound: A new linear-threshold algorithm. Mach. Learn. 2, 285–318 (1987)

    Article  Google Scholar 

  20. Naslund, E., Sawin, W.: Upper bounds for sunflower-free sets, Forum Math. Sigma 5 (2017), Paper No. e15

  21. Pach, J., Agarwal, P.: Combinatorial geometry. Wiley-Interscience, New York (1995)

    Book  MATH  Google Scholar 

  22. Rao, A.: Coding for sunflowers, Discrete Anal. (2020) Paper No. 2

  23. Sauer, N.: On the density of families of sets. J. Combinat. Theory Ser. A 13, 145–147 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  24. Sharir, M.: On \(k\)-sets in arrangements of curves and surfaces. Discrete Comput. Geom. 6, 593–617 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  25. Shelah, S.: A combinatorial problem, stability and order for models and theories in infinitary languages. Pacific J. Math. 41, 247–261 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  26. Shelah, S.: Classification theory and the number of non-isomorphic models. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  27. Smorodinsky, S., Sharir, M.: Selecting points that are heavily covered by pseudo-circles, spheres or rectangles. Combin. Probab. Comput. 13, 389–411 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  28. Spencer, J.: Intersection theorems for systems of sets. Canad. Math. Bull. 20, 249–254 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  29. Vapnik, V., Chervonenkis, A.: On the uniform convergence of relative frequencies of events to their probabilities. Theory Probab. Appl. 16, 264–280 (1971)

    Article  MATH  Google Scholar 

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Acknowledgements

We would like to thank Amir Yehudayoff for suggesting working with the Littlestone dimension, and the SoCG 2021 referees for helpful comments.

Funding

Fox: Supported by a Packard Fellowship and by NSF Awards DMS-1800053 and DMS-2154169. Pach: Supported by NKFIH Grant Nos. K-131529, KKP-133864, Austrian Science Fund Z 342-N31, Ministry of Education and Science of the Russian Federation Mega Grant No. 075-15-2019-1926, ERC Advanced Grant “GeoScape.” Suk: Supported an NSF CAREER award, NSF award DMS-1952786, and an Alfred Sloan Fellowship.

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Fox, J., Pach, J. & Suk, A. Sunflowers in Set Systems of Bounded Dimension. Combinatorica 43, 187–202 (2023). https://doi.org/10.1007/s00493-023-00012-z

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