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The EGH Conjecture and the Sperner property of complete intersections


Authors: Tadahito Harima, Akihito Wachi and Junzo Watanabe
Journal: Proc. Amer. Math. Soc. 145 (2017), 1497-1503
MSC (2010): Primary 13M10; Secondary 13C40
DOI: https://doi.org/10.1090/proc/13347
Published electronically: October 26, 2016
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Abstract: Let $ A$ be a graded complete intersection over a field and $ B$ the monomial complete intersection with the generators of the same degrees as $ A$. The EGH conjecture says that if $ I$ is a graded ideal in $ A$, then there should be an ideal $ J$ in $ B$ such that $ B/J$ and $ A/I$ have the same Hilbert function. We show that if the EGH conjecture is true, then it can be used to prove that every graded complete intersection over any field has the Sperner property.


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Tadahito Harima
Affiliation: Department of Mathematics Education, Niigata University, Niigata, 950-2181 Japan
Email: harima@ed.niigata-u.ac.jp

Akihito Wachi
Affiliation: Department of Mathematics, Hokkaido University of Education, Kushiro, 085-8580 Japan
Email: wachi.akihito@k.hokkyodai.ac.jp

Junzo Watanabe
Affiliation: Department of Mathematics, Tokai University, Hiratsuka, 259-1201 Japan
Email: watanabe.juzno@tokai-u.jp

DOI: https://doi.org/10.1090/proc/13347
Received by editor(s): November 16, 2015
Received by editor(s) in revised form: June 21, 2016
Published electronically: October 26, 2016
Additional Notes: This work was supported by JSPS KAKENHI Grant numbers (C) (15K04812), (C) (23540179).
Communicated by: Irena Peeva
Article copyright: © Copyright 2016 American Mathematical Society