MAT 307 : Combinatorics Lecture 9 - 10 : Extremal combinatorics
نویسنده
چکیده
1 Bipartite forbidden subgraphs We have seen the Erdős-Stone theorem which says that given a forbidden subgraph H, the extremal number of edges is ex(n,H) = 2(1−1/(χ(H)−1)+o(1))n. Here, o(1) means a term tending to zero as n → ∞. This basically resolves the question for forbidden subgraphs H of chromatic number at least 3, since then the answer is roughly cn2 for some constant c > 0. However, for bipartite forbidden subgraphs, χ(H) = 2, this answer is not satisfactory, because we get ex(n,H) = o(n2), which does not determine the order of ex(n,H). Hence, bipartite graphs form the most interesting class of forbidden subgraphs.
منابع مشابه
Topics in Extremal Combinatorics - Notes
3 Lecture 3 8 3.1 Krivelevich’s proof that r(3, k) ≥ ck2/ log k . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Ajtai-Komlós-Szemerédi’s r(3, k) < ck2/ log k . . . . . . . . . . . . . . . . . . . . . . 9 3.3 AKS Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.4 AKS formal proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...
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