Tight Asymptotic Bounds on the Size of Order Graphs

Order graphs were introduced by Arroyuelo et al.\ (TODS 2024) as a generalization of the ring index, to support worst-case optimal joins on relations of arity $d$ while storing as few columns as possible: the index stores one column per edge, so its size is proportional to the number of edges of an order graph. It was known that the minimum number of edges lies between $Ω(2^d\sqrt d)$ and $O(2^d d)$, and the lower bound had been claimed to be $42$ for $d=6$. We first exhibit an order graph with only $40$ edges for $d=6$, and show that the proof of that lower bound holds only for unions of cycles, whereas order graphs can be smaller than the smallest set of cycles. We give a corrected statement and proof that holds for every order graph: there must be at least $\max_m\binom dm\lceil m/2\rceil$ edges. This bound is $40$ for $d=6$, so our $40$-edge graph is optimal. We then settle the asymptotic question: the minimum size is $Θ(2^d\sqrt d)$. This is obtained via an explicit construction, based on symmetric chain decompositions of the Boolean lattice, with at most $(d-1)\binom{d}{\lfloor d/2\rfloor}+2^d$ edges. The upper bound is within a factor $4+O(d^{-1/2})$ of the lower bound, and both numerically and asymptotically better than trie switching.