Description
In this survey talk, we investigate graphs and 3-uniform linear hypergraphs not containing any copy of the cycle $C_5$ in Berge sense (or in general no $C_{2k+1}$).
We study the maximum number of triangles in $C_5$-free graphs, first studied by Bollobas and the speaker, improved later by Alon and Shikhelman and now by Ergemlidze, Methuku, Salia and by the speaker. In $C_{2k+1}$-free graphs it was studied first by Hao Li and the speaker, then it was improved by Furedi an Ozhahya, but here the lower and upper bounds are much farther from each other.
The number of hyperedges in $C_5$-free 3-uniform hypergraphs first was studied by Bollobas and the speaker too, and it was improved by Furedi an Ozhahya. For $C_{2k+1}$-free hypergraphs, the extremal number was estimated by Lemons and the speaker. Recently, we studied it in 3-uniform linear hypergraphs too. Let $F$ be a family of 3-uniform linear hypergraphs.The linear Turan 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. We asymptotically determine this Turan number for $C_5$.
Some other related questions will be discussed too.
(The new results are joint works with Beka Ergemlidze, Abhishek Methuku and partly by Nika Salia)