1Department of Computer Science and Software Engineering, Concordia University, Montréal, Canada. 2Institute of Informatics, University of Gdánsk, Poland. 3Department of Mathematics, University of Idaho - Moscow, Idaho, USA.
The electronic journal of combinatorics, Tome 20 (2013) no. 4
In this paper, we are concerned with calculating $r(k, n)$, the length of the longest $k$-AP free subsequences in $1, 2, \ldots , n$. We prove the basic inequality $r(k, n) \le n − \lfloor m/2\rfloor$, where $n = m(k − 1) + r$ and $r < k − 1$. We also discuss a generalization of a famous conjecture of Szekeres (as appears in Erdős and Turán) and describe a simple greedy algorithm that appears to give an optimal $k$-AP free sequence infinitely often. We provide many exact values of $r(k, n)$ in the Appendix.
Tanbir Ahmed
1
;
Janusz Dybizbánski
2
;
Hunter Snevily
3
1
Department of Computer Science and Software Engineering,
Concordia University, Montréal, Canada.
2
Institute of Informatics,
University of Gdánsk, Poland.
3
Department of Mathematics,
University of Idaho - Moscow, Idaho, USA.
@article{10_37236_3007,
author = {Tanbir Ahmed and Janusz Dybizb\'anski and Hunter Snevily},
title = {Unique sequences containing no \(k\)-term arithmetic progressions},
journal = {The electronic journal of combinatorics},
year = {2013},
volume = {20},
number = {4},
doi = {10.37236/3007},
zbl = {1364.11023},
url = {http://geodesic.mathdoc.fr/articles/10.37236/3007/}
}
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AU - Tanbir Ahmed
AU - Janusz Dybizbánski
AU - Hunter Snevily
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Tanbir Ahmed; Janusz Dybizbánski; Hunter Snevily. Unique sequences containing no \(k\)-term arithmetic progressions. The electronic journal of combinatorics, Tome 20 (2013) no. 4. doi: 10.37236/3007
