A Tight Upper Bound on the Number of Variables for Average-Case k-Clique on Ordered Graphs

Abstract

A first-order sentence ϕ defines k-clique in the average-case if

$$ \lim_{n\to\infty} \Pr_{G = G(n,p)} \big[G \models \varphi \,\Leftrightarrow\, G \text{ contains a $k$-clique}\big] = 1 $$

where G = G(n,p) is the Erdős-Rényi random graph with p (= p(n)) the exact threshold such that $\Pr[G(n,p)$ has a k-clique] = 1/2. We are interested in the question: How many variables are required to define average-case k-clique in first-order logic? Here we consider first-order logic in vocabularies which, in addition to the adjacency relation of G, may include fixed “background” relations on the vertex set {1,…,n} (for example, linear order). Some previous results on this question:

• With no background relations, k/2 variables are necessary and k/2 + O(1) variables are sufficient (Ch. 6 of [7]).

• With arbitrary background relations, k/4 variables are necessary [6].

• With arithmetic background relations (<, +, ×), k/4 + O(1) variables are sufficient (Amano [1]).

In this paper, we tie up a loose end (matching the lower bound of [6] and improving the upper bound of [1]) by showing that k/4 + O(1) variables are sufficient with only a linear order in the background.