Supersaturation Beyond Color-Critical Graphs

The supersaturation problem for a given graph F asks for the minimum number \(h_F(n,q)\) of copies of F in an n-vertex graph with \(\textrm{ex}(n,F)+q\) edges. Subsequent works by Rademacher, Erdős, and Lovász and Simonovits determine the optimal range of q (which is linear in n) for cliques F such that \(h_F(n,q)\) equals the minimum number \(t_F(n,q)\) of copies of F obtained from a maximum F-free n-vertex graph by adding q new edges. A breakthrough result of Mubayi extends this line of research from cliques to color-critical graphs F, and this was further strengthened by Pikhurko and Yilma who established the equality \(h_F(n,q)=t_F(n,q)\) for \(1\le q\le \epsilon _F n\) and sufficiently large n. In this paper, we present several results on the supersaturation problem that extend beyond the existing framework. Firstly, we explicitly construct infinitely many graphs F with restricted properties for which \(h_F(n,q)<q\cdot t_F(n,1)\) holds when \(n\gg q\ge 4\), thus refuting a conjecture of Mubayi. Secondly, we extend the result of Pikhurko–Yilma by showing the equality \(h_F(n,q)=t_F(n,q)\) in the range \(1\le q\le \epsilon _F n\) for any member F in a diverse and abundant graph family (which includes color-critical graphs, disjoint unions of cliques \(K_r\), and the Petersen graph). Lastly, we prove the existence of a graph F for any positive integer s such that \(h_F(n,q)=t_F(n,q)\) holds when \(1\le q\le \epsilon _F n^{1-1/s}\), and \(h_F(n,q)<t_F(n,q)\) when \(n^{1-1/s}/\epsilon _F\le q\le \epsilon _F n\), indicating that \(q=\Theta (n^{1-1/s})\) serves as the threshold for the equality \(h_F(n,q)=t_F(n,q)\). We also discuss some additional remarks and related open problems.