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In 2016, Nath and Sellers proposed a conjecture regarding the precise largestsize of ${(s,ms-1,ms+1)}$-core partitions. In this paper, we prove theirconjecture. One of the key techniques in our proof is to introduce and studythe properties of generalized-$\beta$-sets, which extend the concept of$\beta$-sets for core partitions. Our results can be interpreted as ageneralization of the well-known result of Yang, Zhong, and Zhou concerning thelargest size of $(s,s+1,s+2)$-core partitions.

Proof of Nath-Sellers' Conjecture on the Largest Size of $(s,ms-1,ms+1)$-Core Partitions