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The theories of Baldwin–Shi hypergraphs and their atomic models

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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.

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Correspondence to Danul K. Gunatilleka.

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Partially supported by NSF Grant DMS-1308546.

A Some relevant number theoretic facts

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. 1.

    \(p_k, q_k\) are increasing sequences (and hence \(p_k,q_k\rightarrow \infty \))

  2. 2.

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

  3. 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 \(\lim _{k}p_{2k}-q_{2k}\beta =0\).

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Gunatilleka, D.K. The theories of Baldwin–Shi hypergraphs and their atomic models. Arch. Math. Logic 60, 879–908 (2021).

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