For color-critical F with χ(F)=r+1≥4, λ²(G) ≥ 2(1-1/r)m + q implies N_F(G) ≥ (B_F - o(1)) q m^{(f-2)/2} for small q, with sharp B_F = α_F/4 ⋅ (2r/(r-1))^{f/2}.
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An edge-spectral supersaturation of Mubayi's theorem for color-critical graphs
For color-critical F with χ(F)=r+1≥4, λ²(G) ≥ 2(1-1/r)m + q implies N_F(G) ≥ (B_F - o(1)) q m^{(f-2)/2} for small q, with sharp B_F = α_F/4 ⋅ (2r/(r-1))^{f/2}.