Turán's extremal problem in random graphs: Forbidding odd cycles


For 0<γ≤1 and graphsG andH, we writeG→γH if any γ-proportion of the edges ofG span at least one copy ofH inG. As customary, we writeC k for a cycle of lengthk. We show that, for every fixed integerl≥1 and real η>0, there exists a real constantC=C(l, η), such that almost every random graphG n, p withp=p(n)≥Cn −1+1/2l satisfiesG n,p→1/2+η C 2l+1. In particular, for any fixedl≥1 and η>0, this result implies the existence of very sparse graphsG withG →1/2+η C 2l+1.

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The first author was partially supported by NSERC. The second author was partially supported by FAPESP (Proc. 93/0603-1) and by CNPq (Proc. 300334/93-1). The third author was partially sopported by KBN grant 2 1087 91 01.

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Haxell, P.E., Kohayakawa, Y. & Luczak, T. Turán's extremal problem in random graphs: Forbidding odd cycles. Combinatorica 16, 107–122 (1996). https://doi.org/10.1007/BF01300129

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Mathematics Subject Classification (1991)

  • 05 C 80
  • 05 C 35
  • 05 C 38
  • 05 C 55