We give an upper bound for the rank r of homogeneous (even) Clifford structures on compact manifolds of non-vanishing Euler characteristic. More precisely, we show that if r=2~a·b with b odd, then r≤9 for a=0, r≤10 for a=1, r≤12 for a=2 and r≤16 for a≥3. Moreover, we describe the four limiting cases and show that there is exactly one solution in each case.
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