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We consider the Ising systems in $d$ dimensions with nearest-neighborferromagnetic interactions and long-range repulsive (antiferromagnetic)interactions which decay with a power, $s$, of the distance. The physicalcontext of such models is discussed; primarily this is $d=2$ and $s=3$ where,at long distances, genuine magnetic interactions between genuine magneticdipoles are of this form. We prove that when the power of decay lies above $d$and does not exceed $d+1$, then for all temperatures, the spontaneousmagnetization is zero. In contrast, we also show that for powers exceeding$d+1$ (with $d\ge2$) magnetic order can occur.Comment: 15 pages, CMP style fil

On the Absence of Ferromagnetism in Typical 2D Ferromagnets