F.A.
I have a very picky nearly 4 year old who is gradually improving. The things which have worked are getting her involved in food preparation - she is more willing to try things she has helped make. Also, I get her to try one mouthful of new things but do not make her eat all of it - it does get her to put things in her mouth which she was refusing. I do not let her spit anything out.
Also I just keep putting things on her plate - I don't say anything and I don't make her eat them but I also don't let her take them off her plate. Every so often she will try something e.g recently she took a bite of broccoli having left it a hundred times before. I did not make any kind of fuss - I didn't praise her or anything, but next time I said, well, you ate it last time, and she is gradually eating a little bit more of it.
Also, I find the more I try and persuade or praise the more she refuses - she is quite a stubborn little person and if she sees how much I want her to eat something she quite enjoys the control she wields. So I keep it all as low key as possible and don't let her see how much I want her to eat the veggies. We are slowly getting somewhere!
Also, if she has eaten something OK one time I don't then let her say she doesn't like it next time, and I won't offer a replacement meal, but will keep the one she has refused until she is hungry enough to eat it.
The other thing that helps is slowly adding other flavors to things she likes. Eg she will eat tomato based sauces, so I do the usual sauce but then add a bit of a new herb to it, then a bit more next time. That way she is being exposed to new flavors without necessarily realising.