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Abstract We determine the maximum number of induced copies of a 5‐cycle in a graph on vertices for every . Every extremal construction is a balanced iterated blow‐up of the 5‐cycle with the possible exception of the smallest level where for , the Möbius ladder achieves the same number of induced 5‐cycles as the blow‐up of a 5‐cycle on eight vertices. This result completes the work of Balogh, Hu, Lidický, and Pfender, who proved an asymptotic version of the result. Similarly to their result, we also use the flag algebra method, but we use a new and more sophisticated approach which allows us to extend its use to small graphs. 

Abstract The Erdős–Simonovits stability theorem states that for all ε > 0 there exists α > 0 such that if G is a K r+ 1 -free graph on n vertices with e ( G ) > ex( n , K r +1 )– α n 2 , then one can remove εn 2 edges from G to obtain an r -partite graph. Füredi gave a short proof that one can choose α = ε . We give a bound for the relationship of α and ε which is asymptotically sharp as ε → 0. 

Abstract For a real constant α , let $$\pi _3^\alpha (G)$$ be the minimum of twice the number of K 2 ’s plus α times the number of K 3 ’s over all edge decompositions of G into copies of K 2 and K 3 , where K r denotes the complete graph on r vertices. Let $$\pi _3^\alpha (n)$$ be the maximum of $$\pi _3^\alpha (G)$$ over all graphs G with n vertices. The extremal function $$\pi _3^3(n)$$ was first studied by Győri and Tuza ( Studia Sci. Math. Hungar. 22 (1987) 315–320). In recent progress on this problem, Král’, Lidický, Martins and Pehova ( Combin. Probab. Comput. 28 (2019) 465–472) proved via flag algebras that $$\pi _3^3(n) \le (1/2 + o(1)){n^2}$$ . We extend their result by determining the exact value of $$\pi _3^\alpha (n)$$ and the set of extremal graphs for all α and sufficiently large n . In particular, we show for α = 3 that K n and the complete bipartite graph $${K_{\lfloor n/2 \rfloor,\lceil n/2 \rceil }}$$ are the only possible extremal examples for large n .