  We present new measurements of the temperature-density (T-rho) relation for
neutral hydrogen in the 2.0 < z < 2.8 intergalactic medium (IGM) using a sample
of ~6000 individual HI absorbers fitted with Voigt profiles constrained in all
cases by multiple Lyman series transitions. We find model independent evidence
for a positive correlation between the column density of HI (NHI) and the
minimum observed velocity width of absorbers (bmin). With minimal
interpretation, this implies that the temperature-density relation in the IGM
is not "inverted", contrary to many recent studies. Fitting bmin as a function
of NHI results in line width - column density dependence of the form bdmin =
b_0 [NHI/N_(HI,0)]^(Gamma-1) with a minimum line width at mean density rhobar
[N_(HI, 0) = 10^13.6 cm^-2] of b_0 = 17.9 +- 0.2 km/s and a power-law index of
(Gamma-1) = 0.15 +- 0.02. Using analytic arguments, these measurements imply an
"equation of state" for the IGM at <z>= 2.4 of the form T = T_0
(rho/rhobar)^(gamma-1) with a temperature at mean density of T_0 = (1.94 +-
0.05) x 10^4 K and a power-law index (gamma -1) = 0.46 +- 0.05.
