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We studied analytically and with particle tracking the intrabeam scattering of a high brightness electron beam in a linac. We found that intrabeam scattering plays no significant role in the microbunching instability that develops in linacs for free electron lasers. A partial damping of the instability is envisaged, however, either with dedicated magnetic insertions, or with electron beams of charge density ∼4 times higher than produced by state-of-the-art photoinjectors.