Packing–saturation equivalence conjecture for triangles

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Let Pr(2)P_r(2) be the rr-colour 22-clique packing number, and let satr(K3)\mathrm{sat}_r(K_3) be the corresponding rr-colour saturation parameter. Here f(r)=Θ(g(r))f(r)=\Theta(g(r)) means that f(r)f(r) and g(r)g(r) are within constant factors.

Packing–saturation conjecture.

Pr(2)=Θ(satr(K3)).P_r(2)=\Theta\bigl(\mathrm{sat}_r(K_3)\bigr).

The conjecture concerns whether the two parameters have the same asymptotic order. The source presents it as a more modest open question after discussing possible separation between semisaturation and saturation.

References

Primary source

Yamaan Attwa, Sam Mattheus, Tibor Szabó and Jacques Verstraete, “Improved bounds for the minimum degree of minimal multicolor Ramsey graphs”, arXiv:2510.09068 (2025).

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