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arXiv 1901.01552math.CO
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如何用拉姆齐理论解$K_{2,k}$的Harary问题

How Ramsey theory can be used to solve Harary's problem for $K_{2,k}$

C. J. Jayawardene, C. C. Rousseau, B. Bollobás

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8 pages, 4 figures
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摘要：

Sidorenko和Goddard及Klietman分别证明了Harary猜想$r(C_3,G)\leq 2q+1$（对于任意具有$q$边的孤立自由图G）.在本文中，而不是$C_3$，我们考虑$K_{2,k}$和寻求一个尖锐的上界$r(K_{2,k},G)$在所有图$G$与$q$边。更具体地说，如果是$q\geq 2$，我们将证明$r(C_4,G)\leq kq+1$和等式在$G \cong qK_2$或$K_3$时成立。使用此方法，我们将在$k>2$时将此结果推广到$r(K_{2,k},G)$。我们还将证明，对于每一个图$G$与$q \geq 2$边和没有孤立的顶点，$r(C_4, G) \leq 2p+ q - 2$，其中$p=|V(G)|$和等式成立，如果$G \cong K_3$。

英文摘要：

Harary's conjecture $r(C_3,G)\leq 2q+1$ for every isolated-free graph G with $q$ edges was proved independently by Sidorenko and Goddard and Klietman. In this paper instead of $C_3$ we consider $K_{2,k}$ and seek a sharp upper bound for $r(K_{2,k},G)$ over all graphs $G$ with $q$ edges. More specifically if $q\geq 2$, we will show that $r(C_4,G)\leq kq+1$ and that equality holds if $G \cong qK_2$ or $K_3$. Using this we will generalize this result for $r(K_{2,k},G)$ when $k>2$. We will also show that for every graph $G$ with $q \geq 2$ edges and with no isolated vertices, $r(C_4, G) \leq 2p+ q - 2$ where $p=|V(G)|$ and that equality holds if $G \cong K_3$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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