Bipartite-ness under smooth conditions

Abstract Given a family $\mathcal{F}$ of bipartite graphs, the Zarankiewicz number $z(m,n,\mathcal{F})$ is maximum edges in an $m$ by $n$ graph $G$ that does not contain any member as subgraph (such called -free ). For $1\leq \beta \lt \alpha 2$ , graphs $(\alpha,\beta )$ - smooth if for some $\rho \gt 0$ and every $m\leq n$ $z(m,n,\mathcal{F})=\rho m n^{\alpha -1}+O(n^\beta . Motivated their work on conjecture Erdős Simonovits compactness classic result Andrásfai, Sós, Allen, Keevash, Sudakov Verstraëte proved -smooth there exists $k_0$ such all odd $k\geq k_0$ sufficiently large -vertex $\mathcal{F}\cup \{C_k\}$ with minimum degree at least (\frac{2n}{5}+o(n))^{\alpha -1}$ bipartite. In this paper, we strengthen showing real $\delta Furthermore, our holds under more relaxed notion smoothness, which include families consisting single $K_{s,t}$ when $t\gg s$ We also prove analogous $C_{2\ell }$ $\ell \geq complements Verstraëte.