I'm thinking of buying a new 4k TV, one I have my eyes on only has 3 HDMI 2.0 connections. I read that 2.0 is backwards compatible with 1.4.
Does that mean it will work on my PS3 Slim 250 GB model?
According to this Reddit discussion, HDMI 2.0 ports still support HDMI 1.x cables, so you should be able to use your new TV with your PS3 and it's original HDMI cable.
However, this poster on the Playstation Forums is having difficulty getting his PS3 to recognise the newer HDMI 2.0 cables. A responder posts this:
I couldn't help but notice you have the same TV as me... [snip] I can confirm for you that my ps3 does work with standard HMDI 1.4 cables...
So yes - as long as you continue to use HDMI 1.4 cables, your PS3 will work on a TV that has HDMI 2.0 ports. Your mileage may vary if you try and upgrade to HDMI 2.0 cabling.