Sum-of-Squares and statistical query algorithms cannot efficiently refute or detect multiple disjoint planted cliques or bicliques below the sqrt(n) total-size threshold.
Cryptography from planted graphs: Security with logarithmic-size messages
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Multiple Planted Structures Below $\sqrt{n}$: An SoS Integrality Gap and an SQ Lower Bound
Sum-of-Squares and statistical query algorithms cannot efficiently refute or detect multiple disjoint planted cliques or bicliques below the sqrt(n) total-size threshold.