Specified intersections
Authors:
Dhruv Mubayi and Vojtech Rödl
Journal:
Trans. Amer. Math. Soc. 366 (2014), 491504
MSC (2010):
Primary 05D05
Published electronically:
May 28, 2013
MathSciNet review:
3118403
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Abstract: Let and be a family of subsets of an element set such that for every . Suppose that is the maximum number of consecutive integers contained in and is sufficiently large. Then The first bound complements the previous bound of roughly due to Frankl and the second author which was proved under the assumption that . For , the second bound above becomes better than the first bound. In this case, it yields , and this can be viewed as a generalization (in an asymptotic sense) of the famous Eventown theorem of Berlekamp (1969) and Graver (1975). We conjecture that our bound remains valid as long as . Our second result complements the result of Frankl and the second author in a different direction. Fix and and let . Then, in the notation above, we prove that for sufficiently large, This is essentially sharp, aside from the multiplicative factor of . The short proof uses the FranklWilson theorem and results about the distribution of prime numbers. We conjecture that a similar bound holds for whenever . A similar conjecture when is fixed and is large was made earlier by Frankl (1977) and proved by Frankl and Füredi (1984).
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Additional Information
Dhruv Mubayi
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, Illinois 60607
Email:
mubayi@uic.edu
Vojtech Rödl
Affiliation:
Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322
Email:
rodl@mathcs.emory.edu
DOI:
http://dx.doi.org/10.1090/S000299472013058771
Received by editor(s):
January 1, 1928
Received by editor(s) in revised form:
May 14, 2012, and January 1, 2011
Published electronically:
May 28, 2013
Additional Notes:
The first author’s research was supported in part by NSF grant DMS0969092
The second author’s research was supported in part by NSF grant DMS0800070
Article copyright:
© Copyright 2013
American Mathematical Society
