Hi all,
I have played around this circuit the whole day and after this I came to some conclusions.
1. It doesn't seem to matter if or what condenser you are using. I tried many caps as the condenser. From 0.01uF to 100uF, did not notice any difference at all. Just the bigger capacity cap used to heat up. I got all the same results even if I did not use any condenser at all.
2. It appears that the relay coil resistance does matter. All my relays have a resistance under 100 ohms, that makes them trigger at lower voltages and low voltage at the primary side of the induction coil means also lower voltage at the HV side. I used a small neo magnet to "help" the relay so that more voltage var required for the contacts to close. This made it possible to go from from 5v to 10v with the same current (1A). I have an adjustable power supply so I can adjust the voltage and the current draw. I set the current so that it does not get higher than 1A. And using the help from the magnet, I hot twice the voltage on the primary side and much more brightnes out of the bulb. Here is a picture of the magnet/relay config.



The magnet is placed directly at the relay coil core. If you attach the magnet with the right pole, you will make the relay to trigger at higher voltages, but the relay coil will get hot soon. Also if you use too strong magnets, it will attact the contact through the coil core and close them.

I think that the higher the voltage is across the lightbulb, the brighter the light will be. I will try to find a better induction coil with more secondary windings. Also, has anyone of you already tried tu use the water spark circuit for this