What is the operating temperature of a turbo bypass engine, and how is it kept at that temperature?
Yes, if you could increase the combustion temperature in the hot section of a gas turbine, you could dramatically increase the efficiency of the engine.  The problem of course is finding materials which can sustain structural integrity in these regimes for long service lives and still contain the hot gas; we are currently at the thermal limits of existing materials.

The key problems here are:

Temperatures - gas turbine hot sections operate at 1600°F all the way up to 2600° F for high performance military engines in order to yield high thrust or high useful work for a unit of fuel burned.  No known metals can survive these upper limits and still maintain high yield strengths.  It also causes great thermal expansion problems between non homogenous materials and coatings as well as oxidation and galvanic corrosion.

Stresses - the spools of a gas turbine can spin as fast as 40,000 rpm in the cases of high pressure compressors and high pressure turbines.  This creates tremendous centripetal loads on the blades and other components which they must be able to endure and not exceed the yield strength of the material.

Extended operation between servicing and reliability - the engine must survive these loads for several hundred cycles over a period of several thousand hours between major overhauls.  Redundancy as well as resistance to damage must be built into the design as well.  The consequences are grave; the trustworthiness and reliability of two gas turbines engines are all that separates a successful Seattle to Moscow polar flight in a 777 and 320 people freezing to death in the Arctic Ocean.

Historically gas turbine hot sections have been constructed from alloys of metals with high melting points and good yield strengths like nickel and cobalt of a very high purity.  Titanium and Molybdenum are also beginning to be used in this as well.  Additionally forming techniques such as single crystal casting are used to remove crystal grains from the material to improve its strength.  The blades ('buckets') of a high pressure turbine are also cast with channels in them and have fine holes laser drilled in their surface in order to take compressor bleed air and 'bathe' the boundary layer of the bucket with cooler air, effectively wrapping it in a cool gas blanket to protect it from the high temperature exhaust.

More exotic materials and techniques are being considered for this.  GEAE and P&amp;W both are exploring the use of composite and ceramic blades in hot sections.  These offer high yield strengths at high temperatures but again pose challenges in terms of oxidation and thermal expansion problems.