The theories of Baldwin–Shi hypergraphs and their atomic models

Abstract

We show that the quantifier elimination result for the Shelah-Spencer almost sure theories of sparse random graphs $G(n,n^{-\alpha })$ given by Laskowski (Isr J Math 161:157–186, 2007) extends to their various analogues. The analogues will be obtained as theories of generic structures of certain classes of finite structures with a notion of strong substructure induced by rank functions and we will call the generics Baldwin–Shi hypergraphs. In the process we give a method of constructing extensions whose ‘relative rank’ is negative but arbitrarily small in context. We give a necessary and sufficient condition for the theory of a Baldwin–Shi hypergraph to have atomic models. We further show that for certain well behaved classes of theories of Baldwin–Shi hypergraphs, the existentially closed models and the atomic models correspond.

A Some relevant number theoretic facts

The number theoretic results concerning Diophantine equations can be found in Chapter 5 of [9] and the number theoretic results concerning continued fractions can be found in Chapter 7 therein.

Remark A.1

We note that in the case all the $\overline{\alpha }_E$ are rational the equation $n-{\sum _{E\in L}{\overline{\alpha }_Em_E}}=-\frac{1}{c}$ has infinitely many positive integer solutions, i.e. solutions where n and all of the $m_E$ are positive. This follows from a straightforward argument using basic number theoretic facts regarding the greatest common denominator, the fact that linear diophantine equations have solutions and $c=\frac{\prod _{1\le i\le n}q_i}{\prod _{1\le i\le n}\gcd (q_i,\text {lcm}(q_{i+1},\ldots ,q_n))}$.

Remark A.2

Let $0< \beta < 1$ be irrational. Note that $\beta$ has a simple continued fraction form $[0:a_1,a_2,\ldots ]=0+1\frac{1}{a_1+\frac{1}{a_2+\cdots }}$ where $a_i\in \omega$ is positive for $i\ge 1$. Let $p_k/q_k=[0:a_1,\ldots ,a_k]$ be the simple continued fraction approximation restricted to k-terms. Now:

• 1. $p_k, q_k$ are increasing sequences (and hence $p_k,q_k\rightarrow \infty$)

• 2. $\langle p_{2k}/q_{2k} : k\in \omega \rangle$ is a strictly increasing sequence that converges to $\beta$

• 3. For even k, $\frac{1}{q_{k}(q_{k}+q_{k+1})}< \beta -\frac{p_k}{q_k} < \frac{1}{q_k q_{k+1}}$

Now it follows that $-\frac{1}{q_{2k}}< p_{2k}-q_{2k}\beta < -\frac{1}{q_{2k}+q_{2k+1}}$. This easily yields that $\lim _{k}p_{2k}-q_{2k}\beta =0$.