Many T copies in H-free graphs

Noga Alon, Clara Shikhelman

Research output: Contribution to journalArticlepeer-review

133 Scopus citations


For two graphs T and H with no isolated vertices and for an integer n, let ex(n,T,H) denote the maximum possible number of copies of T in an H-free graph on n vertices. The study of this function when T=K2 is a single edge is the main subject of extremal graph theory. In the present paper we investigate the general function, focusing on the cases of triangles, complete graphs, complete bipartite graphs and trees. These cases reveal several interesting phenomena. Three representative results are: (i) ex(n,K3,C5)≤(1+o(1))32n3/2,(ii) For any fixed m, s≥2m−2 and t≥(s−1)!+1, ex(n,Km,Ks,t)=Θ(nm−(m2)/s), and(iii) For any two trees H and T, ex(n,T,H)=Θ(nm) where m=m(T,H) is an integer depending on H and T (its precise definition is given in Section 1).The first result improves (slightly) an estimate of Bollobás and Győri. The proofs combine combinatorial and probabilistic arguments with simple spectral techniques.

Original languageEnglish (US)
Pages (from-to)146-172
Number of pages27
JournalJournal of Combinatorial Theory. Series B
StatePublished - Nov 1 2016
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics


  • Complete bipartite graphs
  • Complete graphs
  • Extremal graph theory
  • H-free graphs
  • Projective norm graphs


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