About: Tight bounds on the maximum size of a set of permutations with bounded VC-dimension

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rdf:type	skos:Concept http://linked.opendata.cz/ontology/domain/vavai/Vysledek
rdfs:seeAlso	http://dx.doi.org/10.1016/j.jcta.2012.04.004
Description	The VC-dimension of a family P of n-permutations is the largest integer k such that the set of restrictions of the permutations in P on some k-tuple of positions is the set of all k! permutation patterns. Let r(k)(n) be the maximum size of a set of n-permutations with VC-dimension k. Raz showed that r(2)(n) grows exponentially in n. We show that r(3)(n) = 2(Theta(n log alpha (n))) and for every t }= 1, we have r(2t+2) (n) = 2(Theta (n alpha(n)t)) and r(2t+3) (n) = 2 (O(n alpha(n)t log alpha(n))). We also study the maximum number p(k)(n) of 1-entries in an n x n (0.1)-matrix with no (k + 1)-tuple of columns containing all (k + 1)-permutation matrices. We determine that, for example, p(3)(n) = Theta(n alpha(n)) and p(2t+2)(n) = n2((1/t)alpha(n)t +/- O(alpha(n)t-1)) for every t }= 1. We also show that for every positive s there is a slowly growing function zeta(s)(n) (for example zeta(2t+3)(n) = 2(O(alpha t(n))) for every t }= 1) satisfying the following. For all positive integers n and B and every n x n (0, 1)-matrix M with zeta(s)(n)Bn 1-entries, the rows of M can be partitioned into s intervals so that at least B columns contain at least B 1-entries in each of the intervals. The VC-dimension of a family P of n-permutations is the largest integer k such that the set of restrictions of the permutations in P on some k-tuple of positions is the set of all k! permutation patterns. Let r(k)(n) be the maximum size of a set of n-permutations with VC-dimension k. Raz showed that r(2)(n) grows exponentially in n. We show that r(3)(n) = 2(Theta(n log alpha (n))) and for every t }= 1, we have r(2t+2) (n) = 2(Theta (n alpha(n)t)) and r(2t+3) (n) = 2 (O(n alpha(n)t log alpha(n))). We also study the maximum number p(k)(n) of 1-entries in an n x n (0.1)-matrix with no (k + 1)-tuple of columns containing all (k + 1)-permutation matrices. We determine that, for example, p(3)(n) = Theta(n alpha(n)) and p(2t+2)(n) = n2((1/t)alpha(n)t +/- O(alpha(n)t-1)) for every t }= 1. We also show that for every positive s there is a slowly growing function zeta(s)(n) (for example zeta(2t+3)(n) = 2(O(alpha t(n))) for every t }= 1) satisfying the following. For all positive integers n and B and every n x n (0, 1)-matrix M with zeta(s)(n)Bn 1-entries, the rows of M can be partitioned into s intervals so that at least B columns contain at least B 1-entries in each of the intervals. (en)
Title	Tight bounds on the maximum size of a set of permutations with bounded VC-dimension Tight bounds on the maximum size of a set of permutations with bounded VC-dimension (en)
skos:prefLabel	Tight bounds on the maximum size of a set of permutations with bounded VC-dimension Tight bounds on the maximum size of a set of permutations with bounded VC-dimension (en)
skos:notation	RIV/00216208:11320/12:10124696!RIV13-GA0-11320___
http://linked.open...avai/riv/aktivita	P S
http://linked.open...avai/riv/aktivity	P(GBP202/12/G061), S
http://linked.open...iv/cisloPeriodika	7
http://linked.open...vai/riv/dodaniDat	2013
http://linked.open...aciTvurceVysledku	Cibulka, Josef Kynčl, Jan
http://linked.open.../riv/druhVysledku	J - Článek v odborném periodiku
http://linked.open...iv/duvernostUdaju	S - Úplné a pravdivé údaje nepodléhající ochraně podle zvláštních právních předpisů
http://linked.open...titaPredkladatele	Univerzita Karlova v Praze / Matematicko-fyzikální fakulta
http://linked.open...dnocenehoVysledku	174436
http://linked.open...ai/riv/idVysledku	RIV/00216208:11320/12:10124696
http://linked.open...riv/jazykVysledku	eng - angličtina
http://linked.open.../riv/klicovaSlova	Inverse Ackermann function; Set of permutations; Davenport-Schinzel sequence; VC-dimension; Permutation pattern; NONLINEARITY; 0-1 MATRICES; STANLEY-WILF CONJECTURE; DAVENPORT-SCHINZEL SEQUENCES (en)
http://linked.open.../riv/klicoveSlovo	0-1 MATRICES DAVENPORT-SCHINZEL SEQUENCES Davenport-Schinzel sequence Inverse Ackermann function NONLINEARITY Permutation pattern STANLEY-WILF CONJECTURE Set of permutations VC-dimension
http://linked.open...odStatuVydavatele	US - Spojené státy americké
http://linked.open...ontrolniKodProRIV	[B89CD23609B2]
http://linked.open...i/riv/nazevZdroje	JOURNAL OF COMBINATORIAL THEORY SERIES A
http://linked.open...in/vavai/riv/obor	BA
http://linked.open...ichTvurcuVysledku	2 (xsd:int)
http://linked.open...cetTvurcuVysledku	2 (xsd:int)
http://linked.open...vavai/riv/projekt	Center of excellence - Institute for theoretical computer science (CE-ITI)
http://linked.open...UplatneniVysledku	2012
http://linked.open...v/svazekPeriodika	119
http://linked.open...iv/tvurceVysledku	Kynčl, Jan Cibulka, Josef
http://linked.open...ain/vavai/riv/wos	000305820200007
issn	0097-3165
number of pages	18 (xsd:int)
http://bibframe.org/vocab/doi	10.1016/j.jcta.2012.04.004
http://localhost/t...ganizacniJednotka	11320

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