The jaggedness of an order ideal I in a poset P is the number of maximal elements in I plus the number of minimal elements of P not in I . A probability distribution on the set of order ideals of P is toggle-symmetric if for every p ∈ P, the probability that p is maximal in I equals the probability that p is minimal not in I . In this paper, we prove a formula for the expected jaggedness of an order ideal of P under any toggle-symmetric probability distribution when P is the poset of boxes in a skew Young diagram. Our result extends the main combinatorial theorem of Chan-López-Pflueger-Teixidor [Trans. Amer. Math. Soc., forthcoming. 2015, arXiv:1506.00516], who used an expected jaggedness computation as a key ingredient to prove an algebro-geometric formula; and it has applications to homomesies, in the sense of Propp-Roby, of the antichain cardinality statistic for order ideals in partially ordered sets.

We prove that with high probability over the choice of a random graph G from the Erdős-Rényi distribution G(n, 1/2), the n O(d) -time degree d Sum-of-Squares semidefinite programming relaxation for the clique problem will give a value of at least n 1/2−c(d/ log n) 1/2 for some constant c > 0. This yields a nearly tight n 1/2−o(1) bound on the value of this program for any degree d = o(log n). Moreover we introduce a new framework that we call pseudo-calibration to construct Sum of Squares lower bounds. This framework is inspired by taking a computational analog of Bayesian probability theory. It yields a general recipe for constructing good pseudo-distributions (i.e., dual certificates for the Sum-of-Squares semidefinite program), and sheds further light on the ways in which this hierarchy differs from others.

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