Use the colorBlender function from this great post to define a custom gradient that is white for low values and gradually progresses to darker values e.g.
The “Copy to clipboard” button will copy a pure function definition to the clipboard. Paste and set a symbol to its value.
colorFunction =
Blend[{{0.0491,
Directive[RGBColor[1., 0.9999694819562066, 0.9999847409781033],
Opacity[1]]}, {0.1298,
Directive[RGBColor[
0.5867093919279774, 0.47226672770275424`, 0.7917906462195773],
Opacity[1]]}, {0.1767,
Directive[RGBColor[0.250728, 0.225386, 0.769152],
Opacity[1]]}, {0.2043,
Directive[RGBColor[0.24408, 0.361242, 0.816084],
Opacity[1]]}, {0.25,
Directive[RGBColor[0.266122, 0.486664, 0.802529],
Opacity[1]]}, {0.3125,
Directive[RGBColor[0.305919, 0.585575, 0.739666],
Opacity[1]]}, {0.375,
Directive[RGBColor[0.36048, 0.655759, 0.645692],
Opacity[1]]}, {0.4375,
Directive[RGBColor[0.429842, 0.701849, 0.540321],
Opacity[1]]}, {0.5,
Directive[RGBColor[0.513417, 0.72992, 0.440682],
Opacity[1]]}, {0.5625,
Directive[RGBColor[0.607651, 0.743718, 0.358588],
Opacity[1]]}, {0.625,
Directive[RGBColor[0.705038, 0.742591, 0.299167],
Opacity[1]]}, {0.6875,
Directive[RGBColor[0.794549, 0.721158, 0.260829],
Opacity[1]]}, {0.75,
Directive[RGBColor[0.863512, 0.670771, 0.236564],
Opacity[1]]}, {0.8125,
Directive[RGBColor[0.901014, 0.582826, 0.216542],
Opacity[1]]}, {0.875,
Directive[RGBColor[0.902853, 0.453964, 0.192014],
Opacity[1]]}, {0.9375,
Directive[RGBColor[0.878107, 0.293208, 0.160481],
Opacity[1]]}, {1.,
Directive[RGBColor[0.857359, 0.131106, 0.132128],
Opacity[1]]}}, #] &
Use it in the plot
SmoothDensityHistogram[pairs, ColorFunction -> colorFunction,
PlotRange -> {{0, 12}, {0, 12}},
Epilog -> {Point[pairs], Line[{{0, 0}, {25, 25}}]},
MaxRecursion -> 3, ImageSize -> 400]
