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For a graphH    $H$   , letc   (  H  )   =  inf   {   c  :  e   (  G  )   ≥  c  |  G  |impliesG  ≻  H   }$c(H)=\text{inf}\{c:e(G)\ge c|G|\,\,\text{implies}\,\,G\succ H\}$   , whereG  ≻  H    $G\succ H$    means thatH    $H$    is a minor ofG    $G$   . We show that ifH    $H$    has average degreed    $d$   , thenc   (  H  )   ≤   (   0.319    …    +   o  d    (  1  )    )   |  H  |    log  d$c(H)\le (0.319\,\ldots \,+{o}_{d}(1))|H|\sqrt{\mathrm{log}d}$where0.319  …    $0.319\ldots $    is an explicitly defined constant. This bound matches a corresponding lower bound shown to hold for almost all suchH    $H$    by Norin, Reed, Wood and the first author.

On the extremal function for graph minors