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Cross t-Intersecting Integer Sequences from Weighted Erdős–Ko–Rado

Published online by Cambridge University Press:  09 May 2013

NORIHIDE TOKUSHIGE
NORIHIDE TOKUSHIGE*
Affiliation:
College of Education, Ryukyu University, Nishihara, Okinawa 903-0213, Japan (e-mail: [email protected])
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Abstract

Let m,n and t be positive integers. Consider [m]n as the set of sequences of length n on an m-letter alphabet. We say that two subsets A⊂[m]n and B⊂[m]n cross t-intersect if any two sequences a∈A and b∈B match in at least t positions. In this case it is shown that if $m > (1-\frac 1{\sqrt[t]2})^{-1}$ then |A||B|≤(mn−t)2. We derive this result from a weighted version of the Erdős–Ko–Rado theorem concerning cross t-intersecting families of subsets, and we also include the corresponding stability statement. One of our main tools is the eigenvalue method for intersection matrices due to Friedgut [10].

Keywords

Primary 05D05Secondary 05C50
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 4 , July 2013 , pp. 622 - 637
Copyright
Copyright © Cambridge University Press 2013 

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References

[1]Ahlswede, R. and Khachatrian, L. H. (1998) The diametric theorem in Hamming spaces: Optimal anticodes. Adv. Appl. Math. 20 429–449.CrossRefGoogle Scholar
[2]Bey, C. and Engel, K. (2000) Old and new results for the weighted t-intersection problem via AK-methods. In Numbers, Information and Complexity (Althofer, I.et al., eds), Kluwer, pp. 45–74.Google Scholar
[3]Dinur, I. and Safra, S. (2005) On the hardness of approximating minimum vertex cover. Ann. of Math. 162 439–485.Google Scholar
[4]Ellis, D., Friedgut, E. and Pilpel, H. (2011) Intersecting families of permutations. J. Amer. Math. Soc. 24 649–682.Google Scholar
[5]Erdős, P., Ko, C. and Rado, R. (1961) Intersection theorems for systems of finite sets. Quart. J. Math. Oxford (2) 12 313–320.Google Scholar
[6]Frankl, P. and Füredi, Z. (1980) The Erdős–Ko–Rado theorem for integer sequences. SIAM J. Alg. Discrete Math. 1 376–381.CrossRefGoogle Scholar
[7]Frankl, P., Lee, S. J., Siggers, M. and Tokushige, N. An Erdős–Ko–Rado theorem for cross t-intersecting families. arXiv:1303.0657Google Scholar
[8]Frankl, P. and Tokushige, N. (1999) The Erdős–Ko–Rado theorem for integer sequences. Combinatorica 19 55–63.CrossRefGoogle Scholar
[9]Frankl, P. and Wilson, R. M. (1986) The Erdős–Ko–Rado theorem for vector spaces. J. Combin. Theory Ser. A 43 228–236.Google Scholar
[10]Friedgut, E. (2008) On the measure of intersecting families, uniqueness and stability. Combinatorica 28 503–528.Google Scholar
[11]Gromov, M. (2010) Singularities, expanders and topology of maps 2: From combinatorics to topology via algebraic isoperimetry. Geom. Funct. Anal. 20 416–526.CrossRefGoogle Scholar
[12]Kleitman, D. J. (1966) Families of non-disjoint subsets. J. Combin. Theory 1 153–155.Google Scholar
[13]Matsumoto, M. and Tokushige, N. (1989) The exact bound in the Erdős–Ko–Rado theorem for cross-intersecting families. J. Combin. Theory Ser. A 22 90–97.Google Scholar
[14]Tokushige, N. (2005) Intersecting families: Uniform versus weighted. Ryukyu Math. J. 18 89–103.Google Scholar
[15]Tokushige, N.The eigenvalue method for cross t-intersecting families. J. Alg. Combin. DOI: 10.1007/s10801-012-0419-4.Google Scholar
[16]Wilson, R. M. (1984) The exact bound in the Erdős–Ko–Rado theorem. Combinatorica 4 247–257.Google Scholar