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We explore the relationship between active galactic nuclei and star formationin a sample of 513 optically luminous type 1 quasars up to redshifts of $\sim$4hosting extremely high star formation rates (SFRs). The quasars are selected tobe individually detected by the \textit{Herschel} SPIRE instrument at $>$3$\sigma$ at 250 $\mu$m, leading to typical SFRs of order of 1000M$_{\odot}$yr$^{-1}$. We find the average SFRs to increase by almost a factor10 from $z\sim0.5$ to $z\sim3$, mirroring the rise in the comoving SFR densityover the same epoch. However, we find that the SFRs remain approximatelyconstant with increasing accretion luminosity for accretion luminosities above10$^{12}$ L$_{\odot}$. We also find that the SFRs do not correlate with blackhole mass. Both of these results are most plausibly explained by the existenceof a self-regulation process by the starburst at high SFRs, which controls SFRson time-scales comparable to or shorter than the AGN or starburst duty cycles.We additionally find that SFRs do not depend on Eddington ratio at anyredshift, consistent with no relation between SFR and black hole growth rateper unit black hole mass. Finally, we find that high-ionisation broadabsorption line (HiBAL) quasars have indistinguishable far-infrared propertiesto those of classical quasars, consistent with HiBAL quasars being normalquasars observed along a particular line of sight, with the outflows in HiBALquasars not having any measurable effect on the star formation in their hosts.

Extreme star formation events in quasar hosts over ${\bf0.5<\textit{z}<4}$