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

• Benjamin Rossman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7456)

## 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.

## Keywords

Linear Order Random Graph Common Neighbor Adjacency Relation Background Relation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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