Regions of the [m<sub>H</sub> , tan&beta; ] plane excluded in the type-I 2HDM via direct searches for heavy Higgs bosons for (a) cos(&beta;-&alpha;)=-0.1 and (b) cos(&beta;-&alpha;)=+0.1. Observed limits  (coloured filled areas) are quoted at 95% CL. For neutral Higgs bosons, only gluon--gluon fusion is considered for the production mode, with the exception of the result for H/A&rarr; tt&#772;. The hatched region indicates H boson widths larger than 5%, for which the experimental limits for H boson searches are invalid. The cross-sections for the Higgs boson production are calculated using up to NNLO QCD corrections for gluon--gluon fusion as implemented in Sushi [25,26,27,28]. The quoted results for H/A&rarr; tt&#772; assume the heavy Higgs boson to be produced in association with tt&#772; and the cross section for this process is calculated using MadGraph5<sub>A</sub>MC@NLO [34]. Charged Higgs boson production cross-sections are calculated with up to NLO QCD corrections following [35]. All Higgs boson widths and branching ratios have been calculated using 2HDMC [37].