Guest Post on App of Ramsey to Constraint Satisfaction

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(Guest Post by Manuel Bodirsky on a paper of his that applies Ramsey Theory.)

In a recent paper we apply the so-called product Ramsey theorem to classify the computational complexity of a large class of constraint satisfaction problems.

A *temporal constraint language* is a relational structure &Gamma with a first-order definition in (Q,<), the linear order of the rational numbers. The problem CSP(&Gamma) is the following computational problem. Input: A *primitive positive* first-order sentence, that is, a first-order sentence that is built from existential quantifiers and conjunction, but without universal quantification, disjunction, and negation. Question: Is the given sentence true in &Gamma?

In our paper we show that such a problem is NP-complete unless &Gamma is from one out of nice classes where the problem can be solved in polynomial time.

The statement of the product Ramsey theorem that we use is as follows: for all positive integers d, r, m, and k > m, there is a positive integer R such that for all sets S₁,...,S_d of size at least R and an arbitrary coloring of the [m]^d subgrids of S₁ × ... × S_d with r colors, there exists a [k]^d subgrid of S₁ × ... × S_d such that all [m]^d subgrids of the [k]^d subgrid have the same color. (A [k]^d-subgrid of S₁ × ... × S_d is a subset of S₁ × ... × S_d of the form S_1' × ... × S_d', where S_i' is a k-element subset of S_i.)