A generalized Turán problem in random graphs

Wojciech Samotij, Clara Shikhelman

Research output: Contribution to journalArticlepeer-review

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 = 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.

Original languageEnglish
Pages (from-to)283-305
Number of pages23
JournalRandom Structures and Algorithms
Volume56
Issue number2
DOIs
StatePublished - 1 Mar 2020

Keywords

  • Turán's theorem
  • random graphs
  • thresholds

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