三方Zarankiewicz数与范数图
Tripartite Zarankiewicz numbers and norm graphs
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中文总结 AI 辅助
该研究利用Alon等人的商范数图得到三方K_{s,t}-自由图边数的下界，改进Tait等人的上界，恢复t=2时结果并给出K_{3,3}对应的渐近公式，还将结果推广到k分图并应用于渐近确定K_{3,3}的三方多色拉姆齐数。
中文摘要 AI 辅助
对于固定整数s≥t≥2，令ex(n,n,n,K_{s,t})表示各部分含n个顶点的三方K_{s,t}-自由图的最大边数。当s≥(t-1)!+1时，取满足s≥(t-1)!r^{t-1}+1的最大整数r，我们利用Alon、Rónyai和Szabó的商范数图，证明ex(n,n,n,K_{s,t})≥(3/2^{1/t} r^{1-1/t}+o(1))n^{2-1/t}。我们改进了Tait和Timmons的上界，证明对所有s≥t≥2，ex(n,n,n,K_{s,t})≤(3/2^{1/t} (s-t+1)^{1/t}+o(1))n^{2-1/t}。这些界共同恢复了t=2时的结果，并给出新的渐近公式ex(n,n,n,K_{3,3})=(3/∛2+o(1))n^{5/3}。类似结果可推广到不含K_{s,t}且s顶点侧或t顶点侧位于单一部分的k分图。作为三方构造的应用，我们渐近确定了K_{3,3}的三方多色拉姆齐数。
英文摘要
For fixed integers $s\ge t\ge2$, let $\operatorname{ex}(n,n,n,K_{s,t})$ denote the maximum number of edges in a tripartite $K_{s,t}$-free graph with $n$ vertices in each part. When $s\ge(t-1)!+1$, let $r$ be the largest integer satisfying $s\ge(t-1)!r^{t-1}+1$. Using the quotient norm graphs of Alon, Rónyai and Szabó, we prove that \[ \operatorname{ex}(n,n,n,K_{s,t}) \ge \left(\frac{3}{2^{1/t}}r^{1-1/t}+o(1)\right)n^{2-1/t}. \] Improving an upper bound of Tait and Timmons, we prove that, for all $s\ge t\ge 2$, \[ \operatorname{ex}(n,n,n,K_{s,t})\le \left(\frac{3}{2^{1/t}}(s-t+1)^{1/t}+o(1)\right)n^{2-1/t}. \] Together, these bounds recover the results for $t=2$, and give the new asymptotic formula \[ \operatorname{ex}(n,n,n,K_{3,3}) =\left(\frac{3}{\sqrt[3]{2}}+o(1)\right)n^{5/3}. \] Analogous results extend to $k$-partite graphs containing no $K_{s, t}$ whose $s$-vertex or $t$-vertex side lies in a single part. As an application of our tripartite construction, we determine the tripartite multicolor Ramsey number of $K_{3,3}$ asymptotically.
