Sum of squares bounds for the ordering principle

TL;DR

This paper investigates the sum of squares hierarchy's ability to prove the ordering principle, establishing near-tight bounds on the degree needed, which advances understanding of SOS's power in combinatorial proofs.

Contribution

The paper provides tight bounds on the degree of SOS required to prove the ordering principle, showing both an upper bound and a matching lower bound.

Findings

SOS can prove the ordering principle with degree O(√n) log(n).

Any SOS proof requires degree at least Ω(n^{1/2 - ε}) for any ε > 0.

Bounds are nearly tight, characterizing SOS's strength in this context.

Abstract

In this paper, we analyze the sum of squares hierarchy (SOS) on the ordering principle on $n$ elements. We prove that degree $O(\sqrt{n}log(n))$ SOS can prove the ordering principle. We then show that this upper bound is essentially tight by proving that for any $\epsilon > 0$, SOS requires degree $\Omega(n^{\frac{1}{2} - \epsilon})$ to prove the ordering principle on $n$ elements.