<p><b>Abstract</b>—The existence of principal interconnections useful in solving the proliferation problem in higher order Hebbian-type associative memories is introduced. Among all legal interconnections, we prove there exists a subset <it>T</it><sub><it>pr</it></sub> that carries more information than the others. Regardless of the network order <it>p</it>, the elements in <it>T</it><sub><it>pr</it></sub> are shown to be those interconnections <it>T</it> that fall within the range of</p><tf>$$\sqrt {m_s} \le \left| T \right| \le 2 \sqrt {m_s},$$</tf><p>where <it>m</it><sub><it>s</it></sub> equals the number of stored codewords. Memories that use only <it>T</it><sub><it>pr</it></sub> can maintain original generalization performance, using less than 50 percent of the total number of interconnections.</p>