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Noting that d$_{iso}$ > 1 implies that there is no border length b for which µ $^{p}$( b ) is infinite, the previous results can be combined to form the following theorem. 

Theorem. For all graphs with isoperimetric dimension d$_{iso}$ > 1 and vertex borders which are connectable with no more than l · b vertices, a finite critical β$_{c}$ < ∞ exists such that for all β > β$_{c}$ spontaneous magnetization is achieved. 

To the latter category belong the regular lattices in d ≥ 2 dimensions and crystals with any kind of elementary cells; we explicitly note that for an Euclidean lattice in d = 2 dimensions we recover the result by Griffiths[15]. In addition, each vertex border can be connected with no more than l · b vertices in the Sierpinski carpet too, which therefore magnetizes, in accord with the existing literature[23] (see Fig.1). 

Consider now the ladders of infinitely growing height (see Fig.2): they are structures described, at any offset n on the semi-infinite base line, by a non-decreasing integer function h ( n ); as long as the isoperimetric dimension of the ladder is strictly greater than one ( h ( n ) ≥ A$_{0}$n α + B$_{0}$ , for α > 0, A$_{0}$ > 0 and B$_{0}$ > 0) the previous arguments apply, so the ladder magnetizes; on the other hand, when d$_{iso}$ = 1 a little more work is required: the total number V ( b ) of vertices which can be included in at least one border of length b can not grow faster than the number of points on the left of the rightmost border of length b ; the latter is at at offset n ( b ) = max { n $^{′}$| b = h − $^{1}$( n $^{′}$) } , so that V ( b ) satisfies V ( b ) ∼ ∑ n ( b ) i =1 h ( i ) for large b ; when 

for some A$_{0}$, A$_{1}$ > 0, B$_{0}$, B$_{1}$ ≥ 0 and α ≥ 1, the volume satisfies 

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and since the borders are all connected µ ( b ) is exponential and the graph magnetizes. When instead lim$_{n}$$_{→∞}$ h ( n ) log n = 0, for all ϵ > 0 and n large enough h ( n ) ≤ ϵ log n holds; as a consequence, b b 

holds for all ϵ > 0; in this case the sum ∑ $_{b}$µ ( b ) e − 2 β ( b − 1) J$_{min}$ diverges for all temperatures, so the hypotheses of our theorem are not fulfilled. These results are in agreement with a result by Chayes and Chayes[24] about more general structures called d -wedges, where it 

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