We have considered the persistence of unvisited sites of a lattice, i.e., the probability S(t) that a site remains unvisited till time t in the presence of mutually repulsive random walkers in one dimension. The dynamics of this system has direct correspondence to that of the domain walls in a certain system of Ising spins where the number of domain walls becomes fixed following a zero-temperature quench. Here we get the result that S(t) ∝ exp(-αtβ) where β is close to 0.5 and α a function of the density of the walkers ρ. The fraction of persistent sites in the presence of independent walkers of density ρ′ is known to be . We show that a mapping of the interacting walkers' problem to the independent walkers' problem is possible with ρ′ ≤ ρ/(1 - ρ) provided ρ′ and ρ are small. We also discuss some other intricate results obtained in the interacting walkers' case. © 2007 IOP Publishing Ltd.