On Turán problems with bounded fractional matching number

Alon and Frankl [J. Combin. Theory Series B. 165 (2024) 223–229] determined the maximum number of edges in an n -vertex graph with bounded matching and clique numbers. In our first main result, we extend their result to graphs with bounded fractional matching and clique numbers. For our second main result, let F be a fixed graph and let s be an integer. In the final section of their paper, Alon and Frankl initiated the study of the maximum number of edges in n -vertex F -free graphs with matching number at most s . Motivated by this, Gerbner [J. Graph Theory. 106 (2024) 23–29] determined this maximum number of edges, apart from a constant additive term. In our second main result we extend Gerbner's result to n -vertex F -free graphs with fractional matching number at most s . Moreover, we obtain the connected versions of the result of Alon and Frankl and our first result.