This work addresses the block-diagonal semidefinite program (SDP) relaxations for the clique number of the Paley graphs. The size of the maximal clique (clique number) of a graph is a classic NP-complete problem; a Paley graph is a deterministic graph where two vertices are connected if their difference is a quadratic residue (square) in a finite field with the number of elements given by certain primes and prime powers. Improving the upper bound for the Paley graph clique number for prime powers that are non-squares is an open problem in combinatorics. Moreover, since quadratic residues exhibit pseudorandom properties, Paley graphs are related to the construction of deterministic restricted isometries, an open problem in compressed sensing. Recent work provides numerical evidence that the current upper bounds can be improved by the sum-of-squares (SOS) relaxations. In particular, the bounds given by the SOS relaxations of degree 4 (SOS-4) have been empirically observed to be growing at an order smaller than square root of the prime. However, computations of SOS-4 appear to be intractable with respect to large graphs. Gvozdenovic et al. introduced a more computationally efficient block-diagonal hierarchy of SDPs and computed the values of these SDPs of degrees 2 (L2) for the Paley graph clique numbers associated with primes p less or equal to 809, which bound from above the corresponding SOS-4 relaxations. We compute the values of the L2 relaxations for p's between 821 and 997. Our results provide some numerical evidence that these relaxations, and therefore also the SOS-4 relaxations, may be scaling at an order smaller than the square root of p. However, due to the size of the SDPs, we have not been able to compute L2 relaxations for p's greater than 997. Therefore, our scaling estimate is not conclusive and presents an interesting open problem for further study.

翻译：本文探讨了针对Paley图团数的块对角半定规划（SDP）松弛。图的极大团大小（团数）是经典的NP完全问题；Paley图是一种确定性图，其中两个顶点相连当且仅当它们在给定元素数为特定素数或素数幂的有限域中的差值为二次剩余（平方）。改进非平方素数幂的Paley图团数上界是组合数学中的一个开放问题。此外，由于二次剩余具有伪随机性质，Paley图与确定性受限等距的构造相关，后者是压缩感知领域的开放问题。近期研究通过数值证据表明，平方和（SOS）松弛能够改进当前上界。具体而言，4次SOS松弛（SOS-4）给出的上界在经验上以小于素数平方根的阶数增长。然而，针对大图的SOS-4计算似乎难以处理。Gvozdenovic等人引入了一种计算效率更高的SDP块对角层次结构，并计算了素数p ≤ 809时的2次SDP（L2）值，这些值给出了相应SOS-4松弛的上界。我们计算了821 ≤ p ≤ 997范围内的L2松弛值。数值结果初步表明，这些松弛（进而SOS-4松弛）的尺度可能以小于p平方根的阶数增长。然而，由于SDP规模限制，我们未能计算p > 997时的L2松弛值。因此，我们的尺度估计并非定论，并为后续研究提供了一个有趣的开放问题。