Determining the classical Ramsey number \(R(K_5)\) and its variants is notoriously challenging. Even the simplest weakened Ramsey number \(R_2^3(K_5)\) remains unknown, and despite recent breakthroughs by Magnant and Schiermeyer (J Graph Theory 101:455–492, 2022), the Gallai–Ramsey number \(gr_k(K_5)\) is still unresolved. The weakened Gallai–Ramsey number \(gr_s^t(G)\) bridges these two areas. Specifically, a graph H is said to satisfy \(H \xrightarrow {(s,t)} G\) if every t-edge-coloring of H, where each triangle uses at most two colors, forces a subgraph G with its edges colored in at most s colors. The number \(gr_s^t(G)\) is defined as the smallest n such that the complete graph \(K_n\) has this property, while the weakened size Gallai–Ramsey number \(sgr_s^t(G)\) is the minimum number of edges in a graph H with \(H \xrightarrow {(s,t)} G\) . Budden and Wimbish (Australas J Combin 84:375–387, 2022) conjectured that \(gr_2^t(K_5)=2^t+1\) for \(t\ge 3\) , which was fully resolved by Li, Broersma, and Wang (J Graph Theory 101:242–264, 2022). We establish its weakened size counterpart, proving that for \(t\ge 2\) , \(\begin{aligned} sgr_2^t(K_5)=\left( {\begin{array}{c}2^t+1\\ 2\end{array}}\right) . \end{aligned}\)
