Abstract
We study families \({\mathcal {F}}\subseteq 2^{[n]}\) with restricted intersections and prove a conjecture of Snevily in a stronger form for large n. We also obtain stability results for Kleitman’s isodiametric inequality and families with bounded set-wise differences. Our proofs introduce a new twist to the classical linear algebra method, harnessing the non-shadows of \({\mathcal {F}}\), which may be of independent interest.
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References
Alon, N., Babai, L., Suzuki, H.: Multilinear polynomials and Frankl-Ray-Chaudhuri-Wilson type intersection theorems. J. Combin. Theory Ser. A 58(2), 165–180 (1991)
Babai, L.: A short proof of the nonuniform Ray-Chaudhuri-Wilson inequality. Combinatorica 8(1), 133–135 (1988)
Bannai, E., Bannai, E., Stanton, D.: An upper bound for the cardinality of an \(s\)-distance subset in real Euclidean space. II. Combinatorica 3(2), 147–152 (1983)
Blokhuis, A.: A new upper bound for the cardinality of \(2\)-distance sets in Euclidean space. Convexity and graph theory (Jerusalem, 1981). North-Holland Math. Stud. 87, 65–66 (1984)
Bollobás, B.: On generalized graphs. Acta Math. Acad. Sci. Hungar. 16, 447–452 (1965)
Chen, W.Y.C., Liu, J.: Set systems with \(L\)-intersections modulo a prime number. J. Comb. Theory Ser. A 116(1), 120–131 (2009)
Cvetković, D.M.: Chromatic number and the spectrum of a graph. Publ. Inst. Math. (Beograd) (N.S.) 14(28), 25–38 (1972)
Erdős, P., Ko, C., Rado, R.: Intersection theorems for systems of finite sets. Quart. J. Math. Oxford Ser. 2(12), 313–320 (1961)
Fisher, R.A.: An examination of the different possible solutions of a problem in incomplete blocks. Ann. Eugenics 10, 52–75 (1940)
Frankl, P.: Bounding the size of a family knowing the cardinality of differences. Studia Sci. Math. Hungar. 20(1–4), 33–36 (1985)
Frankl, P.: Antichains of fixed diameter. Mosc. J. Comb. Number Theory 7(3), 3–33 (2017)
Frankl, P.: A stability result for families with fixed diameter. Combin. Probab. Comput. 26(4), 506–516 (2017)
Frankl, P.: A stability result for the Katona theorem. J. Comb. Theory Ser. B 122, 869–876 (2017)
Frankl, P., Wilson, R.M.: Intersection theorems with geometric consequences. Combinatorica 1(4), 357–368 (1981)
Gerbner, D., Patkós, B.: Extremal finite set theory. Discrete Mathematics and its Applications. CRC Press, Boca Raton (2019)
Hilton, A.J.W., Milner, E.C.: Some intersection theorems for systems of finite sets. Quart. J. Math. Oxford Ser. 2(18), 369–384 (1967)
Huang, H., Klurman, O., Pohoata, C.: On subsets of the hypercube with prescribed Hamming distances. J. Comb. Theory Ser. A 171, 105156 (2020)
Hwang, K.-W., Kim, Y.: A proof of Alon-Babai-Suzuki’s conjecture and multilinear polynomials. Eur. J. Comb. 43, 289–294 (2015)
Hwang, K.-W., Sheikh, N.N.: Intersection families and Snevily’s conjecture. Eur. J. Comb. 28(3), 843–847 (2007)
Katona, G.: Intersection theorems for systems of finite sets. Acta Math. Acad. Sci. Hungar. 15, 329–337 (1964)
Katona, G.O.H.: Open problems. Matematikus Kurir (1983)
Kleitman, D.J.: On a combinatorial conjecture of Erdős. J. Comb. Theory 1, 209–214 (1966)
Lubell, D.: A short proof of Sperner’s lemma. J. Comb. Theory 1, 299 (1966)
Mešalkin, L.D.: A generalization of Sperner’s theorem on the number of subsets of a finite set. Teor. Verojatnost. i Primenen 8, 219–220 (1963)
Mubayi, D., Zhao, Y.: On the VC-dimension of uniform hypergraphs. J. Algebraic Comb. 25(1), 101–110 (2007)
Ray-Chaudhuri, D.K., Wilson, R.M.: On \(t\)-designs. Osaka Math. J. 12(3), 737–744 (1975)
Snevily, H.S.: On generalizations of the de Bruijn-Erdős theorem. J. Combin. Theory Ser. A 68(1), 232–238 (1994)
Snevily, H.S.: A generalization of the Ray-Chaudhuri-Wilson theorem. J. Combin. Des. 3(5), 349–352 (1995)
Snevily, H.S.: A sharp bound for the number of sets that pairwise intersect at \(k\) positive values. Combinatorica 23(3), 527–533 (2003)
Sperner, E.: Ein Satz über Untermengen einer endlichen Menge. Math. Z. 27(1), 544–548 (1928)
Wang, X., Wei, H., Ge, G.: A strengthened inequality of Alon-Babai-Suzuki’s conjecture on set systems with restricted intersections modulo \(p\). Discrete Math. 341(1), 109–118 (2018)
Yamamoto, K.: Logarithmic order of free distributive lattice. J. Math. Soc. Japan 6, 343–353 (1954)
Acknowledgements
We would like to express our gratitude to the anonymous reviewer for the detailed and constructive comments which are helpful to the improvement of the technical presentation of this paper. We also thank Yongtao Li and Tuan Tran for helpful discussions.
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Gao, J., Liu, H. & Xu, Z. Stability Through Non-Shadows. Combinatorica 43, 1125–1137 (2023). https://doi.org/10.1007/s00493-023-00053-4
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DOI: https://doi.org/10.1007/s00493-023-00053-4