Parameter space a₀ and α. The red region (colour online) in the left is the portion of the parameter space that is discarded by our observations with the most conservative approach that takes in consideration the cumulating effects of space-time fluctuations (see text). The white region in the middle is the additional portion of the parameter space that would be discarded if no compensation due to the cumulating effect of space-time occurs. The grey region in the right is the portion of the parameter space that cannot be excluded in any case. The data obtained with our analysis from the farthest QSOs exclude the random walk and the holographic models of space-time foam in the range of a₀, here reported. These results, indicated by the label QSOs, go far beyond the previous estimates reported in the literature (indicated by z = 5.34 Universe), obtained without considering the cumulating effects of space-time.