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A product involving the $β$-family in stable homotopy theory

In the stable homotopy groups $π_{q(p^n+p^m+1)-3}(S)$ of the sphere spectrum $S$ localized at the prime $p$ greater than three, J. Lin constructed an essential family $ξ_{m,n}$ for $n \geq m + 2 >5$. In this paper, the authors show that the composite $ξ_{m,n}β_{s}\in π_{q(p^n+p^m+sp+s)-5}(S)$ for $2 \leq s < p$ is non-trivial, where $q=2(p-1)$ and $β_s \in π_{q(sp+s-1)-2}(S)$ is the known $β$-family. We show our result by explicit combinatorial analysis of the (modified) May spectral sequence.