The d-dimensional hypercube fails to contain 3-regular expander graphs with about C*2^d/d edges as minors, making the minor-universality threshold of the hypercube exactly of order 2^d/d.
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Tight Bounds for Hypercube Minor-Universality
The d-dimensional hypercube fails to contain 3-regular expander graphs with about C*2^d/d edges as minors, making the minor-universality threshold of the hypercube exactly of order 2^d/d.