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to refuse to change your plans or ideas, especially when someone is trying to persuade you to do so

dig your heels in

to keep doing or talking about the same thing without achieving anything:

go/run round in circles

If two people ... they accept that they have different opinions about something and stop trying to persuade each other that they are right.

agree to differ

to believe something good about someone, rather than something bad, when you have the possibility of doing either:

give sb the benefit of the doubt

a way of dealing with or avoiding a problem:

way round

to go as a group into a public building and refuse to leave or to allow normal activities to continue there until a situation that you are complaining about is changed

sit-in

to admit that you were wrong or that you have been defeated:

back down

something that happens that delays or prevents a process from developing:

setback

a disappointment:

letdown

the rate or act of taking something in/the rate or act of accepting something:

uptake

an act of taking in something, especially breath:

intake

almost certainly:

in all likelihood:

to be important:

be of concern

to finish or destroy something:

put paid to sth

it is not yet certain:

it remains to be seen

the likely future situation:

outlook for

an increase in number or amount:

buildup

(an amount that is) more than is needed:

surplus

a very small piece of fire that flies out from something that is burning, or one that is made by rubbing two hard things together

spark

to exist only in small numbers or amounts:

be thin on the ground

a structure where bees live, either built by people or made by the bees themselves:

hive

a liking or sympathy for someone or something, especially because of shared characteristics/a close similarity between two things:

affinity

to be the right type of person for a particular job or activity (PV)

cut out for something

interesting and exciting arguments:

cut and thrust

learned or succeeded in, and now a part of your experience:

under your belt

anything that can be used to your advantage:

grist to the mill

to criticize something as bad, without value, or unnecessary [formal]

decry

to use a small device attached to a phone in order to listen secretly to what people are saying:

tap

coming or likely to happen very soon:

imminent

the beginning:

the outset

treating everyone fairly and equally:

even-handed

boring and making you feel unhappy:

dreary

almost out of control because of extreme emotion, such as worry:

frantic

continuing in a severe or extreme way:

relentless