Summary: M. Ajtai et al. [J. Comb. Theory, Ser. A 29, 354–360 (1980; Zbl 0455.05045)] proved that for sufficiently large \( t\) every triangle-free graph with \( n\) vertices and average degree \( t\) has an independent set of size at least \( \frac {n}{100t}\log {t}\). We extend this by proving that the number of independent sets in such a graph is at least \[ 2^{\frac{1}{2400}\frac{n}{t}\log^2t}. \] This result is sharp for infinitely many \(t\), \(n\) apart from the constant. An easy consequence of our result is that there exists \(c^\prime>0\) such that every \(n\)-vertex triangle-free graph has at least of independent sets in such a graph is at least \[ 2^{c^\prime\sqrt{n}\log n} \] independent sets. We conjecture that the exponent above can be improved to \(\sqrt{n}(\log n)^{3/2}\). This would be sharp by the celebrated result of J. H. Kim [Random Struct. Algorithms 7, No. 3, 173–207 (1995; Zbl 0832.05084)] which shows that the Ramsey number \(R(3,k)\) has order of magnitude \(k^2/\log k\).