We study how decentralized agents can develop shared vocabularies without global coordination. Answering this question can help us understand the emergence of many communication systems, from bacterial communication to human languages, as well as helping to design algorithms for supporting self-organizing information systems such as social tagging or ad-word systems for the web. We introduce a formal communication model in which senders and receivers can adapt their communicative behaviors through a type of win-stay lose-shift adaptation strategy. We find by simulations and analysis that for a given number of meanings, there exists a threshold for the number of words below which the agents can't converge to a shared vocabulary. Our finding implies that for a communication system to emerge, agents must have the capability of inventing a minimum number of words or sentences. This result also rationalizes the necessity for syntax, as a tool for generating unlimited sentences.