I want some advice, to warn people and to moan.

[Road Rasher]

I paid the seller through PayPal, and the courier I think was PayPal too.

 

[Cuchilo]

If it was pay pal then put in a complaint right now . With both you and the seller shouting blue murder the courier may take notice .
Have you contacted the seller yet ?

 

[GrumpyGregry]

I paid the seller through PayPal, and the courier I think was PayPal too.

Paid them for the purchase of goods? Of sent them the money as a gift?

Edit; damage almost looks like someone stood on the seat stay.

 

[Road Rasher]

The seller set up a PayPal invoice that I paid

 

[TheDoctor]

Wow, lots of nice and not so nice people in here!

I'm currently revising my bike phrases ready for a comprehensive test of all the component names of a bike. So far I've learnt about something called a saddle, a wheel and something called handlebars. I feel I'll be ready for a bike soon.

And when that time comes, I'll make sure it's the Austin Allegro of bikes for no more than £80.

Yeah.
I hear the sarcasm, but I'm not the one with a broken frame, am I?
Seriously, if you don't know exactly what you're doing (and even if you do!) then buying a second-hand bike unseen from Gumtree and getting it couriered by CityLink is a recipe for disaster. As you've discovered.
I'd get in touch with carboncyclerepairs and see what they suggest.
Might be a whole load cheaper than a new frame...