TY - JOUR

T1 - A generalized Turán problem in random graphs

AU - Samotij, Wojciech

AU - Shikhelman, Clara

N1 - Publisher Copyright:
© 2019 Wiley Periodicals, Inc.

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

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AN - SCOPUS:85067393488

VL - 56

SP - 283

EP - 305

JO - Random Structures and Algorithms

JF - Random Structures and Algorithms

SN - 1042-9832

IS - 2

ER -