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In 1994, Alon construct a triangle-free $(n,d,\lambda)$-graph with $d =\Omega(n^{2/3})$ and $\lambda = O(d^{1/2})$ for an exponentially increasingsequence of integers $n$. Using his ingenious construction, we deduce thatthere exist triangle-free $(n,d,\lambda)$-graphs with $d = \Omega(n^{2/3})$ and$\lambda = O( (d \log n)^{1/2} )$ for all sufficiently large $n$.

Dense triangle-free $(n, d, \lambda)$-graphs for all orders