When the giant aliens came lumbering out of the interdimensional fissure, we fought back, building giant robots of our own... which promptly collapsed onto themselves and were unable to move because of (among other things) the square-cube law and lack of materials with adequate tensile strength to stop the damn thing crumpling under its own weight, inertia and torque requirements. We took a step back, mourned our losses, then started using nukes and kinetic missiles on the aliens like sensible people.
A metallurgist studying a crystal of unobtanium diphlebotinide, discovered its unusual property of forming perfectly (down to the near-atomic scale) cylindrical crystals. He found that if he ground the crystal to be within a few nanometres of being exactly 42 times longer in axis than diameter, and waved his hands over it while fumbling his coffee cup, the crystal settled into a phase with (as best they could determine) infinite tensile, compression and shear strength; it is to all intents and purposes indestructible. I believe physically-necessary consequences are an infinite (or at least speed-of-light) speed of sound, and zero coefficient of friction. They subsequently found that if they 'cast' the crystal already having a perfectly cylindrical hole perpendicular to the axis and one-forty-twoth the diameter of the parent cylinder, then it would still set with the hole, allowing them to join crystals together into larger structures.
I think this (long thin spars with the means to fix pinions to transfer force between them) is all you need to use this new material in construction, including superstructures for giant robots. The zero-friction aspect also makes them excellent axles and bearing plates, and I can imagine lots of smart people immediately setting to designing all sorts of complex machines using them (reminds me of childhood construction toys!). But I'm hoping that the nature of the material means they can't be so easily used to create 'armour': you can't weave them or construct any sort of solid plate, for instance, although it could be used as reinforcement behind armour or pressurised enclosures.
Is this material alone enough to 'solve' the standard problems with giant mecha, namely collapsing under their own weight because their legs can't be strong enough, not being able to move their limbs because we can't produce motors with enough torque, and so forth? If not, what problems would remain to overcome?
Let's leave aside the question of why you would choose to attack a giant alien with a giant robot in preference to a nuke even if you could build one. I'm not asking for a mecha that's an appropriate combat weapon, or even humanoid; more like something to compete on a giant version of Robot Wars.