A generalized Turán problem in random graphs

We study the following generalization of the Turan problem in sparse random graphs. Given graphs $T$ and $H$, let $\mathrm{ex}\big(G(n,p), T, H\big)$ be the random variable that counts the largest number of copies of $T$ in a subgraph of $G(n,p)$ that does not contain $H$. We study the threshold phenomena arising in the evolution of the typical value of this random variable, for every $H$ and an arbitrary $2$-balanced $T$. Our results in the case when $m_2(H) > m_2(T)$ are a natural generalization of the Erdős--Stone theorem for $G(n,p)$, which was proved several years ago by Conlon and Gowers and by Schacht; the case $T = K_m$ has been recently resolved by Alon, Kostochka, and Shikhelman. More interestingly, the case when $m_2(H) \le m_2(T)$ exhibits a more complex and subtle behavior. Namely, 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 $\mathrm{ex}\big(G(n,p), T, H\big)$ are given by solutions to deterministic hypergraph Turan-type problems that we are unable to solve in full generality.

[1]  M. Schacht.  Extremal results for random discrete structures , 2016, 1603.00894.

[2]  W. T. Gowers,et al.  On the KŁR conjecture in random graphs , 2013, 1305.2516.

[3]  Ding‐Zhu Du,et al.  Wiley Series in Discrete Mathematics and Optimization , 2014 .

[4]  Wojciech Samotij,et al.  THE METHOD OF HYPERGRAPH CONTAINERS , 2018, Proceedings of the International Congress of Mathematicians (ICM 2018).

[5]  Miklós Simonovits,et al.  Paul Erdős' Influence on Extremal Graph Theory , 2013, The Mathematics of Paul Erdős II.

[6]  Svante Janson,et al.  Random graphs , 2000, Wiley-Interscience series in discrete mathematics and optimization.

[7]  Yoshiharu Kohayakawa,et al.  An Extremal Problem For Random Graphs And The Number Of Graphs With Large Even-Girth , 1998, Comb..

[8]  T. E. Harris.  A lower bound for the critical probability in a certain percolation process , 1960, Mathematical Proceedings of the Cambridge Philosophical Society.

[9]  S. Janson,et al.  Wiley‐Interscience Series in Discrete Mathematics and Optimization , 2011 .

[10]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[11]  Alexandr V. Kostochka,et al.  On independent sets in hypergraphs , 2011, Random Struct. Algorithms.

[12]  Yoshiharu Kohayakawa,et al.  OnK4-free subgraphs of random graphs , 1997, Comb..

[13]  J. Sheehan,et al.  On the number of complete subgraphs contained in certain graphs , 1981, J. Comb. Theory, Ser. B.

[14]  M. Simonovits,et al.  The History of Degenerate (Bipartite) Extremal Graph Problems , 2013, 1306.5167.

[15]  V. Rödl,et al.  Ramsey properties of random discrete structures , 2010 .

[16]  Yoshiharu Kohayakawa,et al.  The chromatic thresholds of graphs , 2011, 1108.1746.

[17]  W. T. Gowers,et al.  Combinatorial theorems in sparse random sets , 2010, 1011.4310.

[18]  Yoshiharu Kohayakawa,et al.  Upper bounds on probability thresholds for asymmetric Ramsey properties , 2014, Random Struct. Algorithms.

[19]  Yoshiharu Kohayakawa,et al.  Turán's Extremal Problem in Random Graphs: Forbidding Even Cycles , 1995, J. Comb. Theory, Ser. B.

[20]  Noga Alon,et al.  Many T copies in H-free graphs , 2015, Electron. Notes Discret. Math..

[21]  D. Saxton,et al.  Hypergraph containers , 2012, 1204.6595.

[22]  Jan Hladký,et al.  On the number of pentagons in triangle-free graphs , 2013, J. Comb. Theory, Ser. A.

[23]  Wojciech Samotij.  Stability results for random discrete structures , 2014, Random Struct. Algorithms.

[24]  Miklós Simonovits,et al.  Supersaturated graphs and hypergraphs , 1983, Comb..

[25]  Danna Zhou,et al.  d. , 1934, Microbial pathogenesis.

[26]  László Babai,et al.  The Mathematics of Paul Erdős I , 2013 .

[27]  Andrzej Grzesik.  On the maximum number of five-cycles in a triangle-free graph , 2012, J. Comb. Theory, Ser. B.

[28]  J. Balogh,et al.  Independent sets in hypergraphs , 2012, 1204.6530.

[29]  P. Erdös.  On the structure of linear graphs , 1946 .

[30]  Yoshiharu Kohayakawa,et al.  Threshold functions for asymmetric Ramsey properties involving cycles , 1997, Random Struct. Algorithms.

[31]  T. Lu.  ON K4-FREE SUBGRAPHS OF RANDOM GRAPHS , 1997 .

[32]  David Saxton,et al.  The number of $C_{2l}$-free graphs , 2013, 1309.2927.

[33]  Y. Person,et al.  SYMMETRIC AND ASYMMETRIC RAMSEY PROPERTIES IN RANDOM HYPERGRAPHS , 2016, Forum of Mathematics, Sigma.

[34]  V. Sós,et al.  On a problem of K. Zarankiewicz , 1954 .

[35]  S. Janson,et al.  Upper tails for subgraph counts in random graphs , 2004 .

[36]  Noga Alon,et al.  Many cliques in $H$-free subgraphs of random graphs , 2016 .