A generalized Turán problem in random graphs

Wojciech Samotij; Clara Shikhelman

doi:10.1002/rsa.20873

A generalized Turán problem in random graphs

Wojciech Samotij^*, Clara Shikhelman

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

We study a generalization of the Turán problem in random graphs. Given graphs T and H, let ex(G(n,p),T,H) be the largest number of copies of T in an H-free subgraph of G(n,p). We study the threshold phenomena arising in the evolution of the typical value of this random variable, for every H and every 2-balanced T. Our results in the case when m₂(H) > m₂(T) are a natural generalization of the Erdős-Stone theorem for G(n,p), proved several years ago by Conlon-Gowers and Schacht; the case T = K_m was previously resolved by Alon, Kostochka, and Shikhelman. The case when m₂(H) ≤ m₂(T) exhibits a more complex behavior. Here, the location(s) of the (possibly multiple) threshold(s) are determined by densities of various coverings of H with copies of T and the typical value(s) of ex(G(n,p),T,H) are given by solutions to deterministic hypergraph Turán-type problems that we are unable to solve in full generality.

Original language	English
Pages (from-to)	283-305
Number of pages	23
Journal	Random Structures and Algorithms
Volume	56
Issue number	2
DOIs	https://doi.org/10.1002/rsa.20873
State	Published - 1 Mar 2020

Funding

Funders	Funder number
German-Israeli Foundation for Scientific Research and Development
Israel Science Foundation	1147/14

Keywords

Turán's theorem
random graphs
thresholds

Access to Document

10.1002/rsa.20873

Cite this

@article{34237294480045aea6f75287d45ea3db,

title = "A generalized Tur{\'a}n problem in random graphs",

abstract = "We study a generalization of the Tur{\'a}n problem in random graphs. Given graphs T and H, let ex(G(n,p),T,H) be the largest number of copies of T in an H-free subgraph of G(n,p). We study the threshold phenomena arising in the evolution of the typical value of this random variable, for every H and every 2-balanced T. Our results in the case when m2(H) > m2(T) are a natural generalization of the Erd{\H o}s-Stone theorem for G(n,p), proved several years ago by Conlon-Gowers and Schacht; the case T = Km was previously resolved by Alon, Kostochka, and Shikhelman. The case when m2(H) ≤ m2(T) exhibits a more complex behavior. Here, the location(s) of the (possibly multiple) threshold(s) are determined by densities of various coverings of H with copies of T and the typical value(s) of ex(G(n,p),T,H) are given by solutions to deterministic hypergraph Tur{\'a}n-type problems that we are unable to solve in full generality.",

keywords = "Tur{\'a}n's theorem, random graphs, thresholds",

author = "Wojciech Samotij and Clara Shikhelman",

note = "Publisher Copyright: {\textcopyright} 2019 Wiley Periodicals, Inc.",

year = "2020",

month = mar,

day = "1",

doi = "10.1002/rsa.20873",

language = "אנגלית",

volume = "56",

pages = "283--305",

journal = "Random Structures and Algorithms",

issn = "1042-9832",

publisher = "John Wiley and Sons Ltd",

number = "2",

}

TY - JOUR

T1 - A generalized Turán problem in random graphs

AU - Samotij, Wojciech

AU - Shikhelman, Clara

PY - 2020/3/1

Y1 - 2020/3/1

N2 - We study a generalization of the Turán problem in random graphs. Given graphs T and H, let ex(G(n,p),T,H) be the largest number of copies of T in an H-free subgraph of G(n,p). We study the threshold phenomena arising in the evolution of the typical value of this random variable, for every H and every 2-balanced T. Our results in the case when m2(H) > m2(T) are a natural generalization of the Erdős-Stone theorem for G(n,p), proved several years ago by Conlon-Gowers and Schacht; the case T = Km was previously resolved by Alon, Kostochka, and Shikhelman. The case when m2(H) ≤ m2(T) exhibits a more complex behavior. Here, the location(s) of the (possibly multiple) threshold(s) are determined by densities of various coverings of H with copies of T and the typical value(s) of ex(G(n,p),T,H) are given by solutions to deterministic hypergraph Turán-type problems that we are unable to solve in full generality.

AB - We study a generalization of the Turán problem in random graphs. Given graphs T and H, let ex(G(n,p),T,H) be the largest number of copies of T in an H-free subgraph of G(n,p). We study the threshold phenomena arising in the evolution of the typical value of this random variable, for every H and every 2-balanced T. Our results in the case when m2(H) > m2(T) are a natural generalization of the Erdős-Stone theorem for G(n,p), proved several years ago by Conlon-Gowers and Schacht; the case T = Km was previously resolved by Alon, Kostochka, and Shikhelman. The case when m2(H) ≤ m2(T) exhibits a more complex behavior. Here, the location(s) of the (possibly multiple) threshold(s) are determined by densities of various coverings of H with copies of T and the typical value(s) of ex(G(n,p),T,H) are given by solutions to deterministic hypergraph Turán-type problems that we are unable to solve in full generality.

KW - Turán's theorem

KW - random graphs

KW - thresholds

UR - http://www.scopus.com/inward/record.url?scp=85067393488&partnerID=8YFLogxK

U2 - 10.1002/rsa.20873

DO - 10.1002/rsa.20873

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:85067393488

SN - 1042-9832

VL - 56

SP - 283

EP - 305

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

IS - 2

ER -

A generalized Turán problem in random graphs

Abstract

Funding

Keywords

Access to Document

Other files and links

Fingerprint

Cite this