Miklós–Manickam–Singhi conjectures on partial geometries
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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.
KeywordsPartial geometries MMS conjecture EKR theorem
Mathematics Subject ClassificationPrimary 05B25 51E20 05E30
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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