A product involving the β-family in stable homotopy theory

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