Each input bit is independently deleted without an erasure marker; the exact capacity is unknown for every nontrivial deletion probability.
Open
\(0.1221(1-d)<C_{\mathrm{del}}(d)\le0.3578(1-d)\)
Channel capacities and open gaps.
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The capacity region of this binary-input broadcast channel remains unknown.
Open
\(\mathcal R_{\mathrm{Marton}}\subseteq\mathcal C_{\mathrm{BSSC}}\subseteq\mathcal R_{\mathrm{UV}}\)
A causal relay assists a source, but decode-forward and the cut-set bound do not coincide in general.
Open
\(R_{\mathrm{DF}}\le C\le R_{\mathrm{cut}}\)
The capacity region for two arbitrary broadcast receivers remains unknown outside important ordered subclasses.
Open
\(\mathcal R_{\mathrm{Marton}}\subseteq\mathcal C_{\mathrm{BC}}\subseteq\mathcal R_{\mathrm{UV}}\)
Two transmitter-receiver pairs interfere, and the exact capacity region is unknown in general.
Open
\(\mathcal R_{\mathrm{HK}}\subseteq\mathcal C_{\mathrm{IC}}\subseteq\mathcal R_{\mathrm{outer}}\)
A relay observes a noisy channel output and sends information over a separate noiseless finite-capacity link, but capacity is unknown in general.
Open
\(R_{\mathrm{CF}}\le C(R_0)\le C_{\mathrm{cut}}\)
The Shannon capacity of the seven-cycle remains unknown.
Open
\(367^{1/5}\le\Theta(C_7)\le\frac{7\cos(\pi/7)}{1+\cos(\pi/7)}\)
The linear-encoder symmetric capacity is known, while unrestricted nonlinear capacity remains open.
Open
\(\frac5{13}\le C_{\mathrm{sym}}\le\frac{11}{28}\)
A six-message, ten-receiver groupcast instance has linear-encoder capacity 5/13 and a non-Shannon nonlinear upper bound 11/28.
Open
\(\frac5{13}\le C_{\mathrm{sym}}\le\frac{11}{28}\)
For a finite confusability graph, the regularized independence number defines capacity but is difficult to compute or characterize.
Open
\(\Theta(G)=\sup_{n\ge1}\alpha(G^{\boxtimes n})^{1/n}\)