The $$\ell $$ l -exclusion problem is to design an algorithm which guarantees that up to $$\ell $$ l processes and no more may simultaneously access identical copies of the same non-sharable resource when there are several competing processes. For $$\ell =1$$ l = 1 , the 1-exclusion problem is the familiar mutual exclusion problem. The simplest deadlock-free algorithm for mutual exclusion requires only one single-writer non-atomic bit per process (Burns in SIGACT News 10(2):42–47, 1978; Burns and Lynch in Inf Comput 107(2):171–184, 1993; Lamport in J ACM 33:327–348, 1986). This algorithm is known to be space optimal (Burns and Lynch in 18th Annual Allerton conference on communication, control and computing, pp 833–842, 1980; Burns and Lynch in Inf Comput 107(2):171–184, 1993). For over 20 years now it has remained an intriguing open problem whether a similar type of algorithm, which uses only one single-writer bit per process, exists also for $$\ell $$ l -exclusion for some $$\ell \ge 2$$ l ? 2 . We resolve this longstanding open problem. For any $$\ell $$ l and $$n$$ n , we provide a tight space bound on the number of single-writer bits required to solve $$\ell $$ l -exclusion for $$n$$ n processes. It follows from our results that it is not possible to solve $$\ell $$ l -exclusion with one single-writer bit per process, for any $$\ell \ge 2$$ l ? 2 . In an attempt to understand the inherent difference between the space complexity of mutual exclusion and that of $$\ell $$ l -exclusion for $$\ell \ge 2$$ l ? 2 , we define a weaker version of $$\ell $$ l -exclusion in which the liveness property is relaxed, and show that, similarly to mutual exclusion, this weaker version can be solved using one single-writer non-atomic bit per process.