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

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F12%3A10124696" target="_blank" >RIV/00216208:11320/12:10124696 - isvavai.cz</a>

Výsledek na webu

<a href="http://dx.doi.org/10.1016/j.jcta.2012.04.004" target="_blank" >http://dx.doi.org/10.1016/j.jcta.2012.04.004</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.jcta.2012.04.004" target="_blank" >10.1016/j.jcta.2012.04.004</a>

Jazyk výsledku

angličtina

Název v původním jazyce

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

Popis výsledku v původním jazyce

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

Název v anglickém jazyce

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

Popis výsledku anglicky

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

Druh

J_x - Nezařazeno - Článek v odborném periodiku (Jimp, Jsc a Jost)

CEP obor

BA - Obecná matematika

OECD FORD obor

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Projekt

<a href="/cs/project/GBP202%2F12%2FG061" target="_blank" >GBP202/12/G061: Centrum excelence - Institut teoretické informatiky (CE-ITI)</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)<br>S - Specificky vyzkum na vysokych skolach

Rok uplatnění

2012

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

JOURNAL OF COMBINATORIAL THEORY SERIES A

ISSN

0097-3165

e-ISSN

—

Svazek periodika

119

Číslo periodika v rámci svazku

7

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

18

Strana od-do

1461-1478

Kód UT WoS článku

000305820200007

EID výsledku v databázi Scopus

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