Creative thinking has long been associated with spreading of activation through concepts within semantic memory. Despite its theoretical importance, little is yet known about how semantic memory structure facilitates and constrains idea production. We examine one potential influence on spreading activation during divergent thinking known as the fan effect: increasing knowledge about a concept leads to increasing interference from conceptually related information. Specifically, we tested whether cue association size—an index of semantic richness reflecting the average number of elements associated with a given concept—impacts the quantity (fluency) and quality (originality) of responses generated during the alternate uses task (AUT). We hypothesized that low-association AUT cues should benefit originality at the cost of fluency because such cues are embedded within a semantic network with fewer conceptual elements, thus yielding lesser interference from closely-related concepts. This hypothesis was confirmed in three experiments. Furthermore, we found an interaction with individual differences in fluid intelligence in the low-association AUT cues, suggesting that constraints of sparse semantic knowledge can be overcome with top-down intervention. The findings indicate that semantic richness differentially impacts the quality and quantity of generated ideas, and that cognitive control processes can facilitate idea production when conceptual knowledge is limited.