Let F be a family of 3-uniform linear hypergraphs. The linear Turán number of F is the maximum possible number of edges in a 3-uniform linear hypergraph on n vertices which contains no member of F as a subhypergraph.In this paper we show that the linear Turán number of the five cycle C 5 (in the Berge sense) is 1 3 √ 3 n 3/2 asymptotically. We also show that the linear Turán number of the four cycle C 4 and {C 3 , C 4 } are equal asmptotically, which is a strengthening of a theorem of Lazebnik and Verstraëte [17].We establish a connection between the linear Turán number of the linear cycle of length 2k + 1 and the extremal number of edges in a graph of girth more than 2k − 2. Combining our result and a theorem of Collier-Cartaino, Graber and Jiang [8], we obtain that the linear Turán number of the linear cycle of length 2k + 1 is Θ(n 1+ 1 k ) for k = 2, 3, 4, 6.

We prove that the maximum number of triangles in a C 5‐free graph on n vertices is at most 122(1+o(1))n3∕2, improving an estimate of Alon and Shikhelman.

We introduce a new approach and prove that the maximum number of triangles in a C 5 -free graph on n vertices is at mostWe also show a connection to r-uniform hypergraphs without (Berge) cycles of length less than six, and estimate their maximum possible size.

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Let F be a fixed graph. The rainbow Turán number of F is defined as the maximum number of edges in a graph on n vertices that has a proper edge-coloring with no rainbow copy of F (where a rainbow copy of F means a copy of F all of whose edges have different colours). The systematic study of such problems was initiated by Keevash, Mubayi, Sudakov and Verstraëte.In this paper, we show that the rainbow Turán number of a path with k + 1 edges is less than Ä 9k 7 + 2 ä n, improving an earlier estimate of Johnston, Palmer and Sarkar.