Building on recent work of Mattheus and Verstraëte, we establish a general connection between Ramsey numbers of the form $r(F,t)$ for $F$ a fixed graph and a variant of the Zarankiewicz problem asking for the maximum number of 1s in an $m$ by $n$ $0/1$-matrix that does not have any matrix from a fixed finite family $\mathcal{L}(F)$ derived from $F$ as a submatrix. As an application, we give new lower bounds for the Ramsey numbers $r(C_5,t)$ and $r(C_7,t)$, namely, $r(C_5,t)=\stackrel{\sim }{\mathrm{\Omega }}(t^{\frac{10}{7}})$ and $r(C_7,t)=\stackrel{\sim }{\mathrm{\Omega }}(t^{\frac{5}{4}})$. We also show how the truth of a plausible conjecture about Zarankiewicz numbers would allow an approximate determination of $r(C_{2\ell +1},t)$ for any fixed integer $\ell ⩾2$.

中文翻译：

---

拉姆齐数和扎兰凯维奇问题

基于 Mattheus 和 Verstraëte 最近的工作，我们在形式为 Ramsey 数之间建立了一般联系$r（F,t）$为了$F$固定图和 Zarankiewicz 问题的变体，要求一个图中 1 的最大数量$米$经过$n$ $0/1$- 不具有来自固定有限族的任何矩阵的矩阵$\mathcal{L}（F）$源自$F$作为子矩阵。作为一个应用，我们为拉姆齐数给出了新的下限$r（C_5,t）$和$r（C_7,t）$，即$r（C_5,t）=\stackrel{～}{\mathrm{\Omega }}（t^{\frac{10}{7}}）$和$r（C_7,t）=\stackrel{～}{\mathrm{\Omega }}（t^{\frac{5}{4}}）$。我们还展示了关于扎兰凯维奇数的合理猜想的真实性如何允许近似确定$r（C_{2\ell +1},t）$对于任何固定整数$\ell ⩾2$。