Measurements of the t-t and p-t coincidence events in the $^3$H ($alpha$, ttp) reaction have been obtained at $E_alpha$ incident energy of 67.2 MeV. Various appropriate angular configurations of detectors were chosen in order to observe the population of the $^6$He$^*$ state at around 18 MeV. Its contribution appears at the $E_{rm tt}$ relative energy of 6.0 MeV by the analysis of bidimensional spectra. We found the formation of the $^6$He excited state at $E^* = 18.3 pm 0.2$ MeV (with a $Gamma$ width of 1.1 $pm$ 0.3 MeV) by the decay into the t+t binary channel, since the threshold energy of the t+t channel is 12.31 MeV. In each analyzed bidimensional energy spectrum of ($E_{rm t}$, $E_{rm t}$) and ($E_{rm p}$, $E_{rm t}$) coincidence events resonance structures are present due to the formation of both $^6$He$^*$ and $^4$He$^*$ excited states. Our results on the $E^*$ and $Gamma$ values regarding the $^6$He$^*$ level of about 18 MeV are compared with the results obtained by other reactions. Moreover, we also found new $Gamma$ width values of 0.7 $pm$ 0.3 and 0.8 $pm$ 0.4 MeV for the 14.0 $pm$ 0.4 and 16.1 $pm$ 0.4 MeV $^6$He levels, respectively.
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