AI 中文总结
本文否证了Parczyk和Spiegel关于Schur三元组彩虹着色最大比例的猜想，将渐近界改进为介于9/22和8/15之间，并推广到k-着色情形。
AI 中文摘要
Parczyk和Spiegel开创了对Schur定理的反Ramsey重数变体的研究，并证明了在$\{1,\dots,n\}$的$3$-着色中，Schur三元组可以被彩虹着色的最大比例渐近地介于$0.4$和$0.66364$之间。此外，他们猜想其下界是最优的。我们否证了这一猜想并证明了新的界。特别地，我们证明了在$\{1,\dots,n\}$的$3$-着色中，彩虹Schur三元组的最大比例渐近地介于$9/22$和$8/15$之间。此外，我们研究了更一般的$k$-着色情形下的问题，并建立了新的非平凡界。
英文摘要
Parczyk and Spiegel initiated the study of an anti-Ramsey multiplicity variant of Schur's theorem and proved that the maximum fraction of Schur triples that can be rainbow in a $3$-coloring of $\{ 1, \dots ,n \}$ is bounded asymptotically between $0.4$ and $0.66364$. Furthermore, they conjectured that their lower bound is optimal. We disprove this conjecture and prove new bounds. In particular, we show that the maximum fraction of rainbow Schur triples that can be rainbow in a $3$-coloring of $\{ 1, \dots ,n \}$ lies between $9/22$ and $8/15$ asymptotically. Moreover, we study the problem in the general $k$-color setting and establish new non-trivial bounds.
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