Clique immersions and independence number

The analogue of Hadwiger’s conjecture for the immersion order states that every graph G contains Kχ(G) as an immersion. If true, this would imply with n vertices and independence number α K⌈nα⌉ best currently known bound is due to Gauthier, Le Wollan, who recently proved a clique on ⌈χ(G)−43.54⌉ vertices. Their result implies n-vertex ⌈n3.54α−1.13⌉ We improve all α≥3, by showing α≥3 ⌊n2.25α−f(α)⌋−1 vertices, where f nonnegative function.