Let $\mathcal{X}= \{1,-1\}^n$, $E$ the set of edges and $J$ some real-valued matrix. Van der Waerden's theorem gives constant $c$ and set of edge weights such that

$$\sum_\mathbf{x\in \mathcal{X}} \exp \sum_{ij \in E} J_{ij} x_i x_j = c \sum_{A\in C} f(A) $$

Where $C$ consists of Eulerian subgraphs over $E$, $f(A)$ is a products of $A$'s edge weights.

Edge weights are strictly below 1 in magnitude, so one could approximate the sum by only considering Eulerian subgraphs that include shortest loops. However, a small number of such loops can give a large number of Eulerian subgraphs. Is there a more efficient approach?

Let $\mathcal{X}= \{1,-1\}^n$, $E$ the set of edges and $J$ some real-valued matrix. Van der Waerden's theorem gives constant $c$ and set of edge weights such that

$$\sum_\mathbf{x\in \mathcal{X}} \exp \sum_{ij \in E} J_{ij} x_i x_j = c \sum_{A\in C} f(A) $$

Where $C$ consists of Eulerian subgraphs over $E$, $f(A)$ is the weight of $A$ defined as the product of weights of edges in $A$

Edge weights are strictly below 1 in magnitude. Suppose only a small number of self-avoiding loops on the graph have non-negligible weight. We can approximate the sum by only considering Eulerian subgraphs including these loops. However, a small number of such loops can give a large number of Eulerian subgraphs. Is there a more efficient way?