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The edges of $G$ can be colored with at most $\\frac{3}{2}\\Delta$ colors by Shannon's theorem. We study lower bounds on the size of subgraphs of $G$ that can be colored with $\\Delta$ colors.\n  Shannon's Theorem gives a bound of $\\frac{\\Delta}{\\lfloor\\frac{3}{2}\\Delta\\rfloor}|E|$. However, for $\\Delta=3$, Kami\\'{n}ski and Kowalik [SWAT'10] showed that there is a 3-edge-colorable subgraph of size at least $\\frac{7}{9}|E|$, unless $G$ has a connected component isomorphic to $K_3+e$ (a $K_3$ with an arbitrary edge doubled). 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