Abstract
In this paper we give a proof of the Miklós–Manickam–Singhi (MMS) conjecture for some partial geometries. Specifically, we give a condition on partial geometries which implies that the MMS conjecture holds. Further, several specific partial geometries that are counterexamples to the conjecture are described.
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05 February 2019
The original version of this article unfortunately contained a mistake. The authors incorrectly quoted Manickam-Mikl?s-Singhi (MMS) conjecture as ?Mikl?s?Manickam?Singhi?. The correct term for the conjecture is Manickam?Mikl?s?Singhi.
05 February 2019
The original version of this article unfortunately contained a mistake. The authors incorrectly quoted Manickam-Mikl��s-Singhi (MMS) conjecture as ���Mikl��s���Manickam���Singhi���. The correct term for the conjecture is Manickam���Mikl��s���Singhi.
References
Alon N., Huang H., Sudakov B.: Nonnegative \(k\)-sums, fractional covers, and probability of small deviations. J. Comb. Theory Ser. B 102(3), 784–796 (2012).
Bier T.: A distribution invariant for association schemes and strongly regular graphs. Linear Algebra Appl. 57, 105–113 (1984).
Bier T., Delsarte P.: Some bounds for the distribution numbers of an association scheme. Eur. J. Comb. 9(1), 1–5 (1988).
Brouwer A.E., Cohen A.M., Neumaier A.: Distance-Regular Graphs, vol. 18. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer, Berlin (1989).
Cameron P.J., van Lint J.H.: On the partial geometry \({\rm pg}(6,\,6,\,2)\). J. Comb. Theory Ser. A 32(2), 252–255 (1982).
Chowdhury A., Sarkis G., Shahriari S.: The Manickam-Miklós-Singhi conjectures for sets and vector spaces. J. Comb. Theory Ser. A 128, 84–103 (2014).
Colbourn C.J., Dinitz J.H.: Handbook of Combinatorial Designs, 2nd edn. Chapman & Hall/CRC, Boca Raton, FL (2007).
De Clerck F., Van Maldeghem H.: Some classes of rank 2 geometries, Chap. 2. In: Buekenhout F. (ed.) Handbook of Incidence Geometry, pp. 434–471. Elsevier, North Holland (1995).
De Clerck F.: Partial and semipartial geometries: an update. Discret. Math. 267(1–3), 75–86 (2003). Combinatorics 2000 (Gaeta).
De Clerck F., Delanote M., Hamilton N., Mathon R.: Perp-systems and partial geometries. Adv. Geom. 2(1), 1–12 (2002).
Doyen J., Wilson R.M.: Embeddings of Steiner triple systems. Discret. Math. 5, 229–239 (1973).
Erdős P., Ko C., Rado R.: Intersection theorems for systems of finite sets. Q. J. Math. Oxford Ser. 2(12), 313–320 (1961).
Godsil C., Meagher K.: Erdős-Ko-Rado Theorems: Algebraic Approaches. Cambridge Studies in Advanced Mathematics, vol. 149. Cambridge University Press, Cambridge (2016).
Haemers W.: A new partial geometry constructed from the Hoffman-Singleton graph. Finite Geom. Designs 49, 119–127 (1981).
Huang H., Sudakov B.: The minimum number of nonnegative edges in hypergraphs. Electron. J. Comb. (2014). arXiv:1309.2549
Jungnickel D., Tonchev V.D.: The number of designs with geometric parameters grows exponentially. Des. Codes Cryptogr. 55(2–3), 131–140 (2010).
Kaski P., Östergård P.R.J., Topalova S., Zlatarski R.: Steiner triple systems of order 19 and 21 with subsystems of order 7. Discret. Math. 308(13), 2732–2741 (2008).
MacNeish H.F.: Euler squares. Ann. Math. (2) 23(3), 221–227 (1922).
Manickam N., Miklós D.: On the number of nonnegative partial sums of a nonnegative sum. In: Combinatorics (Eger, 1987) Colloq. Math. Soc. János Bolyai, vol. 52, pp. 385–392. North-Holland, Amsterdam (1988)
Manickam N., Singhi N.M.: First distribution invariants and EKR theorems. J. Comb. Theory Ser. A 48(1), 91–103 (1988).
Pokrovskiy A.: A linear bound on the Manickam-Miklós-Singhi conjecture. J. Comb. Theory Ser. A 133, 280–306 (2015).
van Lint J.H., Schrijver A.: Construction of strongly regular graphs, two-weight codes and partial geometries by finite fields. Combinatorica 1(1), 63–73 (1981).
Witt E.: Die 5-fach transitiven Gruppen von Mathieu. Abh. Math. Sem. Univ. Hamburg 12(1), 256–264 (1937).
Acknowledgements
The authors would like to thank Klaus Metsch for pointing out the construction by Jungnickel and Tonchev used in Lemma 4.5. The authors would also like to thank Ameera Chowdhury for discussing MMS conjectures with them and providing various preprints of her work. The authors would like to thank John Bamberg for his suggestion to include a discussion of all known partial geometries with \(\alpha =2\). The authors would like to thank the referees for their very helpful and constructive comments on the presentation of the results. Research supported in part by an NSERC Discovery Research Grant, Application No.: RGPIN-341214-2013. The first author acknowledges support from a PIMS Postdoctoral Fellowship.
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Communicated by K. Metsch.
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Ihringer, F., Meagher, K. Miklós–Manickam–Singhi conjectures on partial geometries. Des. Codes Cryptogr. 86, 1311–1327 (2018). https://doi.org/10.1007/s10623-017-0397-6
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DOI: https://doi.org/10.1007/s10623-017-0397-6