On the extremal graphs in generalized Turan problems

Given two graphs H and F , the generalized Tur & aacute;n number ex ( n , H , F ) is the largest number of copies of H in an n -vertex F -free graph. For every graph F , we present an extremal graph for a generalized Tur & aacute;n problem. More precisely, we present a graph H not containing F and for sufficiently large n we present an F -free n -vertex graph G that contains exactly ex ( n , H , F ) copies of H . (c) 2024 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY -NC license (http://creativecommons .org /licenses /by-nc /4 .0/).