A note on the equivalence of Motzskin's maximal density and Ruzsa's measures of intersectivity
Main Article Content
Abstract
In this short note, we see the equivalence of Motzkin's maximal density of integral sets whose no two elements are allowed to differ by an element of a given set $M$ of positive integers and the measures of difference intersectivity defined by Ruzsa. Further more, the maximal density $\mu(M)$has been determined for some infinite sets $M$ and in a specific case of generalized arithmetic progression of dimension two a lower bound has been given for $\mu(M)$.
Article Details
How to Cite
Pandey, R.
(2014).
A note on the equivalence of Motzskin's maximal density and Ruzsa's measures of intersectivity.
Acta Mathematica Universitatis Comenianae, 83(2), 157-163.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/70/17
Issue
Section
Articles
References
[1] D. G. Cantor and B. Gordon, Sequences of integers with missing dierences, J. Combin. Theory Ser. A, 14(1973), 281-287.
[2] G. J. Chang, L. Huang and X. Zhu, The circular chromatic numbers and the fractional chromatic numbers of distance graphs, European J. Combin., 19(1998), 423-431.
[3] G. Chang, D. Liu and X. Zhu, Distance Graphs and T-colorings, J. Combin. Theory Ser. B, 75(1999), 159-169.
[4] R. B. Eggleton, P. Erdos and D. K. Skilton, Coloring the Real Line, J. Combin. Theory Ser. B, 39(1985), 86-100.
[5] J. R. Griggs and D.D.F. Liu, The Channel Assignment Problem for Mutually Adjacent Sites, J. Combin. Theory Ser. A, 68(1994), 169-183.
[6] S. Gupta, Sets of integers with missing dierences, J. Combin. Theory Ser. A, 89(2000), 55-69.
[7] S. Gupta and A. Tripathi, Density of M-sets in Arithmetic Progression, Acta Arith., 89.3(1999), 255-257.
[8] N. M. Haralambis, Sets of Integers with Missing Dierences, J. Combin. Theory Ser. A, 23(1977), 22-33.
[9] A. Kemnitz and H. Kolberg, Coloring of Integer Distance Graphs, Discrete Math., 191(1998), 11-123.
[10] D.D.-F. Liu and X. Zhu, Fractional Chromatic Number for Distance Graphs with Large Clique Size, J. Graph Theory, 47(2004), 129-146.
[11] D.D.-F. Liu and X. Zhu, Fractional Chromatic Number for Distance Graphs generated by two-interval sets, European J. Combin., 29(7)(2008), 1733-1742.
[12] T. S. Motzkin, Unpublished problem collection.
[13] R. K. Pandey, A. Tripathi, On the density of integral sets with missing dierences, Integers, Vol.9 Supp(2009), Article 12.
[14] R. K. Pandey, A. Tripathi, On the density of integral sets with missing dierences from sets related to arithmetic progressions, J. Number Theory, 131(2011), 634-647.
[15] R. K. Pandey, A. Tripathi, A note on a problem of Motzkin regarding density of integral sets with missing dierences, J. Integer Sequences, 14(2011), Article 11.6.3.
[16] J. H. Rabinowitz and V. K. Proulax, An asymptotic approach to the channel assignment problem, SIAM J. Alg. Disc. Methods, 6(1985), 507-518.
[17] I. Z. Ruzsa, On measures of intersectivity, Acta Math. Hungar., 43(1984), 335-340.
[18] A. Sarkozy, On dierence sets of sequences of integers, I, Acta Math. Acad. Sci. Hungar., 31(1978), 125-149.
[19] A. Sarkozy, On dierence sets of sequences of integers, II, Ann. Univ. Sci. Budapest. Etvs. Sect. Math. , 21(1978), 45-53.
[20] A. Sarkozy, On dierence sets of sequences of integers, III, Acta Math. Acad. Sci. Hungar., 31(1978), 355-386.
[2] G. J. Chang, L. Huang and X. Zhu, The circular chromatic numbers and the fractional chromatic numbers of distance graphs, European J. Combin., 19(1998), 423-431.
[3] G. Chang, D. Liu and X. Zhu, Distance Graphs and T-colorings, J. Combin. Theory Ser. B, 75(1999), 159-169.
[4] R. B. Eggleton, P. Erdos and D. K. Skilton, Coloring the Real Line, J. Combin. Theory Ser. B, 39(1985), 86-100.
[5] J. R. Griggs and D.D.F. Liu, The Channel Assignment Problem for Mutually Adjacent Sites, J. Combin. Theory Ser. A, 68(1994), 169-183.
[6] S. Gupta, Sets of integers with missing dierences, J. Combin. Theory Ser. A, 89(2000), 55-69.
[7] S. Gupta and A. Tripathi, Density of M-sets in Arithmetic Progression, Acta Arith., 89.3(1999), 255-257.
[8] N. M. Haralambis, Sets of Integers with Missing Dierences, J. Combin. Theory Ser. A, 23(1977), 22-33.
[9] A. Kemnitz and H. Kolberg, Coloring of Integer Distance Graphs, Discrete Math., 191(1998), 11-123.
[10] D.D.-F. Liu and X. Zhu, Fractional Chromatic Number for Distance Graphs with Large Clique Size, J. Graph Theory, 47(2004), 129-146.
[11] D.D.-F. Liu and X. Zhu, Fractional Chromatic Number for Distance Graphs generated by two-interval sets, European J. Combin., 29(7)(2008), 1733-1742.
[12] T. S. Motzkin, Unpublished problem collection.
[13] R. K. Pandey, A. Tripathi, On the density of integral sets with missing dierences, Integers, Vol.9 Supp(2009), Article 12.
[14] R. K. Pandey, A. Tripathi, On the density of integral sets with missing dierences from sets related to arithmetic progressions, J. Number Theory, 131(2011), 634-647.
[15] R. K. Pandey, A. Tripathi, A note on a problem of Motzkin regarding density of integral sets with missing dierences, J. Integer Sequences, 14(2011), Article 11.6.3.
[16] J. H. Rabinowitz and V. K. Proulax, An asymptotic approach to the channel assignment problem, SIAM J. Alg. Disc. Methods, 6(1985), 507-518.
[17] I. Z. Ruzsa, On measures of intersectivity, Acta Math. Hungar., 43(1984), 335-340.
[18] A. Sarkozy, On dierence sets of sequences of integers, I, Acta Math. Acad. Sci. Hungar., 31(1978), 125-149.
[19] A. Sarkozy, On dierence sets of sequences of integers, II, Ann. Univ. Sci. Budapest. Etvs. Sect. Math. , 21(1978), 45-53.
[20] A. Sarkozy, On dierence sets of sequences of integers, III, Acta Math. Acad. Sci. Hungar., 31(1978), 355-386.