I think that your own “No Kids Night” would work quite well. Kind of like the way places use “Mom’s Day Out” on their ads, you could say something like “No Kid’s Night! Parents’ Night Off” or some such.
Or, you might say “Only Bring Kids Whom You Want to be Pepper Sprayed”. You get the idea.
" Oh, hey friend! Just a heads up: we’re looking at hosting a dinner party soon that we’d love you to come to. I’m never sure how much notice is needed to organise sitters - if we gave you two weeks warning, would that generally be enough?"
Assuming I’m not being whooshed …
Most people are afraid to speak clearly and unambiguously. The OP is all about a person who wants us to guess at what some other people think because he’s too embarrassed to ask them what they think. He’s afraid that somehow the act of asking will give offense. And much of the advice others have given the OP amount to passive aggressive ways to possibly send a message that, if noticed, would tend to cause the recipient to probably act the way the OP wants. Maybe.
To me, that’s doing things the hard way. The very hard and very unreliable way. The easy and reliable way is to speak plain polite English. Ask simple questions which have simple unambiguous answers. And act on the belief that the person who answered told you the truth as they understand it.
In fewer words than the above, but more than my original aphorism: Use Plain polite English in social situations. It’s not offensive, it avoids more misunderstandings than it creates, and it’s quick and reliable. It’s much easier than most people fear it is.
Just say so.
That said, many SNPs, especially the mothers of autistic children, often think that the child wasn’t invited because, well, because.
For years we had a big “invite everyone and then they invite their friends” sort of houseparty. When people started having kids, the kids came. We had kids, they had kids and it was great.
You know, until THOSE kids ruined it for everyone. The kids that - unsupervised by their parents - got into boxes in the basement that hadn’t been unpacked since we’d moved in a decade before and strew things all over the basement. Or perhaps not THOSE kids, THOSE parents, the ones who don’t watch their kids and/or haven’t taught them not to dig through boxes in someone else’s basement.
So the next year we changed the invitations to say “an adult evening of food, grown up beverages and grown up conversation without children” - then got rid of our own kids for the night. And THOSE parents didn’t show up - they can’t get a sitter for their precious snowflakes.
As an adult who is not a parent, but has lots of friends who are, I wouldn’t make that assumption. My friends’ kids are invited to events by default when we host parties at our house. They were invited to our wedding, too.
That wording is as clear and non-confrontational as I can imagine. I would be surprised if it would cause offence (I wouldn’t be offended, and I have kids), while at the same time it would be unimaginable that anyone could read it and think that it’s OK to bring kids.
I’m afraid I’ve never been a supporter of apart-height.
You were being whooshed. 
Reminds me of White Noise:
“You’re going to die.”
“Give it to me straight, doc, none of this medical mumbo jumbo!”
My beloved mother was only 5’ 2" so you would have to careful of what you put down . LOL! I do love your idea.
On a side note - I’ve gotten frustrated that when my wifes relations are in town they deem it necessary to bring their 2 big dogs over to our house when they visit.
I remember back when I was single people thought nothing about bringing their kids over when clearly my house wasnt child proof and I had my action figures on display.
Wise words.
When my son was about 8 and we were living in Egypt, we were invited to the wedding of a dear childhood friend, in Washington. We would have gone as we were going to be in the US anyway on the date of the wedding, but there was a very pointed “no children may attend” on the invitation, and I had no idea how we were going to arrange for childcare while flying in and out of DC just for the wedding. So, that was that - we just couldn’t go, although I was very disappointed.
Well, duh. Action figures are for kids!
Uninvited kids are bad, but at least vaguely understandable “we wanted to come and couldn’t get/afford a sitter” is at least flattering.
I’ve had the uninvited dog thing. Cat chasing dogs when I’ve had an ill cat. Specifically excluded dogs where we found out through the grapevine that they were bringing them anyway for a week long stay and I had to play the bad guy and say “no dogs or find somewhere else to stay” - at that point I had two very small children, an elderly cat, and really didn’t need two sort of trained large dogs in my house.
Clear communication is the key. In this case, it isn’t that the kids aren’t “welcome” in your home, it’s that the purpose of the party is to give parents some time away from the kids. “Parent’s Night Out” is a good way to phrase it.
But then you have to decide which parent gets to go.
You’re really on a roll, aren’t you? 
If someone has a big, friendly backyard wedding and wants kids there that is great! I still wouldn’t bring my kid but I totally get that other people would.
IMO weddings are on a short list I have of places kids shouldn’t go because they are adult in nature. Weddings, black tie events, places of employment that don’t cater to the public (like large office buildings as opposed to restaurants) are all places I wouldn’t take a young child because it isn’t fair to the grown ups there. Funerals, graduations, etc. are the kind of thing I wouldn’t take a young child to because it isn’t fair to the child to expect them to be quiet and still through the solemn event and not disturb anyone else. I know lots of people who would take their children to any of those places or events and they aren’t wrong but they are making choices I wouldn’t. However, people who throw a tantrum over not bringing their kids to one of those events when specifically asked not to are a special kind of wrong that should be punishable by catapault.
This confuses me. I am not trying to be difficult. On one hand, you think that if you take your kid to a wedding–even if invited–it’s not being fair to the adults there, but you don’t think other people who make the same choice are wrong? So their kid isn’t unfair to the adults there, but your kid is?
We need a code word for “stop, now you’re just being obnoxious about it!” :smack: