Odd independence number of even-dimensional hypercubes

Determine the odd independence number of the even-dimensional hypercube $Q_d$, or improve the known upper and lower bounds; in particular, establish or refute whether $\sid(Q_d)/2^d$ converges to $1/2$ as even $d$ tends to infinity.

Background

For odd-dimensional hypercubes, the paper proves $\sid(Q_d)=2^{d-1}$. For even-dimensional hypercubes, it establishes an upper bound from the regular-graph argument and several lower bounds, including exact values for Q4Q_4, Q6Q_6, and Q8Q_8. The authors explicitly state that the tightness of the even-dimensional upper bound is unknown and formulate the exact-value and asymptotic questions as an open problem.

References

We do not know, however, whether or not the upper bound for the case of even $d$ is tight.

— The odd independence number of graphs, I: Foundations and classical classes  (2509.20763 - Caro et al., 25 Sep 2025) in Section 4, “Application to hypercubes,” immediately before Proposition 4.1 and Section 7, “Concluding remarks and open problems”