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Wouldn’t use distemper. Too fragile for modern family living. More for retired couple in a 1500s cottage vibe. Little greene is great in general. anything really decent these days is About 100 per 5l. . Dulux Trade is fine though and the cheapest I would go tbh. Heard good things about tikkarilla recently too. Damp - are the skirtings rotten behind ? Get a damp meter in there . Hallways often bad as lack heating and upvc door seals limit ventilation. I don’t buy the damp injection thing but hacking off the plaster and doing a hard cement render skirt would potentially work - but also may push the issue elsewhere- depends if it’s happening because of a breached damp proof course . In a house if that age it would been slate - which may have cracked. We had big issues with similar in a house at the bottom of Highgate hill . My theory was the ground / surface water came down and sat around our foundations. Fuck knows tho damp is such a headache to diagnose .
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Flaking is not necessarily damp - it could be efflorescence from the brick.
Fixing what is causing the damp is more important than paint choice though - even a clay / chalk paint is going to mitigate damp a small bit.
The fact that it's a UPVC door suggests that (if it's a solid wall construction), whoever did the last redecoration didn't care for period correct materials, and - I'm going to guess wildly - skimmed everything in gypsum plaster. In which case, paint with hammerite if you care to - the gypsum isn't breathable at all.
To the damp: there's often obvious causes: bad gutters, bad render, breached DPC (can be internal or external), missing caulking, splashback, inadequate drainage / broken drainage.
Is there evidence of damp?
Earthborn clay paint is great for allowing walls to “breath”. It’s not the cheapest.