Note on induced paths in sparse random graphs

We show that for $d\ge d_0(ε)$, with high probability, the random graph $G(n,d/n)$ contains an induced path of length $(3/2-ε)\frac{n}{d}\log d$. This improves a result obtained independently by Luczak and Suen in the early 90s, and answers a question of Fernandez de la Vega. Along the way, we generalize a recent result of Cooley, Draganić, Kang and Sudakov who studied the analogous problem for induced matchings.

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