Chapter50A Ramsey Theory Essay: The Dinner Party

Project by: Ryan J. Coghill.

$\textbf{Summary}\text{:}$ I wrote about Ramsey's theorem and the Dinner Party problem, because I found the imagery of an example to be very helpful to my own understanding but wanted to connect it with the proof. To visually connect it with Ramsey's theorem, I made each paragraph a mini Dinner Party example. Just like between people I have three traits:

• Share line length.
• Share a rhyme.
• Share a bold thematic element.

If you share a trait, same class, same age, same favourite ice cream flavour, for example, you are in some way similar.

Ramsey said: any six people three have met each other or three are mutual strangers. Why not say there are three which have something in common. Having met is something in common.

$\textbf{A Ramsey Theory Essay: The Dinner Party by Ryan J. Coghill}$

The Ramsey house stood aloft on the hill.
Unassuming perhaps from the others,
But this was a very special house,
For it was here that lived Frank Ramsey
Who had found, he thought, a very clever mathematical truth,
And he was preparing to reveal it at his Dinner Party that evening.

He'd written and sent out imploring letters
To his colleagues, his students and peers.
For he would need six to show what he had in mind.
And it was at six that they arrived upon his door.
Studious fellows and mathematicians,
Eager to see the brilliant proof Ramsey had in store.

“Welcome all” cried Ramsey his face aglow as they came in.
First Richard entered and he in turn smiled with his host.
Next Bertel walked through the door deep in theory and thought.
And Issai with enthusiasm joined them.
Followed by Ron who was sharing his own theorems with Issai.
And Paul, the last of the great masters in the house

It was not long that Paul and Ron did wait to ask his news.
They were eager to discover the reason he had called upon them.
$$We must know why you've called on us in such haste,
Dear Friend,$''$ they begged him $$we simply can not wait!
What is this math that you seem to have found?$''$

“Why dear Friends it is here, You?re the math, Look around.”
$$Why the six of us are not here by random,
Though jointly one might rightly claim we look the part.
It is in fact in the randomness that allows me to show you something truly beautiful.
I myself see five dear friends, but some of you are yet to meet.
And I have found that it must be so such that in any room are a three.
Who know each other completely or three who jointly did never greet.$"$

“Why yes,” said Ron $$its true indeed,
For I have known Paul often and Richard too
And I am sure they work together often.
So you have found your trio as you wished,
But in any random grouping is absurd without a proof.
I find that rather difficult to believe.$"$

“But you see here” stated Ramsey, $$I've gathered you all six today
- And I myself have taken it on to let you eye up this fact.
Under the watchful eye, some things here match and some misalign.
My first thought adjunct and these last two have rhymes.
What we need is a mapping, with nodes for each thought,
And when I draw in the edges, your eyes will not deceive.$"$

While he drew up his argumentative six lines of thought his guests flocked around him.
A set of crimson red lines joined some and ocean blue for others.
He joined up each mention of similar taste and soon found a triangle all in one shade.
Thrice had he used his imagery of eyes, which arose a commotion in one of his peers.
“You've simply stated the same thing thrice” cried Issai “What of those with none such trios.”
“Perhaps in my haste sir, I have, but it is true, and this certainly is not the only case it appears,”

$$Why even in your stuporous state I can demonstrate this pattern.
If you wish to give a case of some six nodes without these monochromatic links,
By all means I offer you my pad and pens,
And six nodes as you have seen me do here.
And take some time and you will find but only.
Monochromatic triangles are guaranteed to appear.$"$

And Issai took his pad from him and sat and noted down.
And mumbled to himself while he worked amongst the graphs.
“You'll see,” stated Ramsey, $$I'm certain he'll find no counter.
It is a matter that even pigeons can understand.$"$
And surely after several minutes Issai rose $$You have convinced me in part.
But I secede by example alone, what matter do pigeons play in this.$"$

$$Take the roof of the towering steeple where one will often find some pigeons.
They make it their home there and in amongst the slats.
By my count last night as I walked up along the path,
There were two holes in which a bird may roost to spend the night.
And yet if you had looked this morning as I did you would have spied five pigeons there.$"$
And his guest saw his point “you mean to say some holes were shared.”

$$When you take this graph with six nodes and join together every edge
It seems so clear that each must join with five others,
And just as there were pigeons on the church
With only two colouring pens, red and blue,
Certainly it must be that each comes more than once,
And perhaps you see the next step of the link.$"$

“Why” saw Bertel “it must have been three times marked in red!”
“Or perhaps” chimed Paul $$only two, and three in blue,
And maybe four and one or five and naught, but always at least one at least is three.$"$
“And of those two of those three coloured edges” prompted Ramsey $$is it not so,
That they connect to two new nodes, shall we call them A and B,
And A must join with B with some other colour else a triangle is formed.$"$

That made sense to all his guests, and he pressed them on some more
“There is a third edge at least of the same colour which leads to C”
By only that same logic A must join C as it did before
But now what of the edge joining B and C? What colour shall it be?
If red, we form a triangle with A and blue with original point
Always a match and we find monochromatic shapes to be a guarantee.

“Now sit down for some supper and let us enjoy ourselves.”
But Richard sat there puzzled for a moment or two,
And as they ate, he queried the Table “Why must we have six nodes?”
$$Why, my friend it merely follows from the proof.
We require five connections to force the three,
And with lesser the nodes we can say nothing of the sort.$"$

“What of seven nodes? Surely” cried Richard “the shape is there as well.”
$$Why certainly. The six can be within the larger set fixed for any set so big.
But these are special, these six nodes, as it is the smallest of such numbers.
With which to force three or three of either colour connections,
And this is where we ask what of fours, and fives, and sixes?
what of n for as big I can take it? Can I find of that shape all colour connections?$"$

$$It surely is the case that it can be done!
And now I offer this chance out to each of you.
A future in the mathematics of this field
I have found a marvelous notion to wield.
Many have found it before, I am sure.
But I show it to you now so that many more will find it soon.$"$

$$My sirs I am sure that what I say you believe
When I say this is true it can be shown to be
Not for a guess. In fact the triangle is a certainty
And I can not speak of what size a quad would force or more,
Only that its sure to be shown if we take enough.
This is chaos complete in its inevitable order.$"$