7 Draft Poems

At Marfreley’s Bar in Houston, Texas

In a dim lit mural behind the bar,
two swans amble in front of a plantation:
its white house lies against the river, lonely

for the cover of more trees that the artist
left out, as the rushing water
empties into the dark dandelion breeze

of rewritten histories. And I had wanted to see
a single woman out, tonight, sitting
alone, like me at the bar, looking

at their life, the plantation, the swans swallowing
small sips of whatever they find in front
of themselves, any parts of a life that might

make sense, tell me I have done the right things.

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Ghazal to Disquietude

1

Drowned in the honk-squeal above the guard rail, I can almost hear
waves sweep in as the soft susurration in the tip of their lips melts the sand between my curled toes.

Noise is now everywhere I want to be
without it. Cars swoosh past Galveston beach roaring their inept monstrous lungs. I can barely

breathe. Or think. Why do trees and blades of every green thing shudder?
Because we are a hyper-intelligent insidious poison? Cats and dogs cling to us in shock and awe.

Ninety-five percent of a car’s energy goes towards moving simply itself not the passengers.
Or rather that’s 2,500 pounds of wastefulness before the crux of tissue steering the steel.

In Hermann Memorial Park a yellow-blue finch tries to sing and fails
in the roar and wall of sound the cars shed in their wake on the I-10 adjoining the beige greenery.

I nod off under a canker tree. A whale whistles out of its water fountain, breathing.
I roll under such plushness, floating with barnacles and sticky ambergris. So glued are our dream’s illogical logic.

I am a sticky carbuncle tearing through the earth’s thin breathability. It’s afternoon in Houston.
I shower again. I scrunch into a starched shirt. I rope my throat with a dead worm’s shiny excrement.

2

My right ear is dead. When I was three
German measles like dappled freckles grew in me

killing the nerve. Now, left ear still good, I hear pretty well
the unprettiness in my parents voices as they divorce:

the light fades as I listen in, on the mosquito bitten dark
roof above the living room window, then roll on my back

to swallow my insignificance in the drifting milky way above.
Now the frogs have started up. A few ducks quack. A splash

might be catfish come to nibble at the stars
tangled in cheap tabloid floating on the pond’s scum.

Pain makes a squelch in my chest like tossed gravel
settling into decay layer at ponds pitch black bottom.

———————————-

While sipping coffee, I read what one student wrote:
“The surviving fifty rare whooping cranes
with their seven-foot wingspread that propels them
in their annual migration from northern Canada
to the Gulf of Mexico fly unerringly and
swiftly overhead as they migrate southward
using a kind of built-in radar
in their search for winter quarters
near Aransas Pass.”

Surviving fifty myself, feeling rare and whooping
with my six-foot slouch that propels me nowhere
in my daily migrations from the kitchen to the couch,
I live by the Gulf of Mexico, sleep unerringly and
swiftly, undercover, my dreams migrate southward
using a kind of built-in slinky
in search for vaginal quarters
near my wife’s Aransas Pass.

To be surviving melanoma is rare
with its seven wretched drugs I puke, that propels
me out of the gothic hospital to monthly migrations of chemo;
swimming in the Gulf of Mexico, on my back, I float unerringly and
slowly, overheard, the nurses’ whispers migrate southward
out of memory, which is a kind of built-in shit-breeder
when I am in pain and searching for the way out
near the dark rings of Uranus.

But survival is everything rare as whooping
or her pubic hair spread to propel me
in my daily migrations from her coffer to wherever
it is in the Gulf of Mexico I am off to, I unerringly
admit to caring enough to love her butt
less than I ought too as I migrate southward
using a kind of built-in stupidity
in my blindly succumbing to what is expected of me
clearly perfecting it into a fairly fucked life.

———————————————————

Litany

See the purple and green crayon alphabet scrawled on yellow sticky notes stapled to tiny Glen Hills cardboard orange juice containers sucked empty by a strawberry-headed freckled girl named Melissa Alexander Winsum,

See the cardboard, folded and wax coated, that once held the orange juice within it, was wood that came from somewhere green and quiet with squirrels that stretched out on the upholding limbs sucking towards the sun their green certitude of elm or pine or oak,

See how Melissa tied together her carton creation with thick pink fuzzy wool string pulled through holes in the juice containers pricked with a three-fifths whittled down number two Venus pencil she over sharpened while working excited in Miss Thurstin’s after school art class last Tuesday,

See how the wool string grew out of a sheep’s skin, that then kept it warm through a snowy Spring, how that wool sprouted, cell upon cell, a protein made from the very grass the sheep was grazing on, from x-ray sun to chlorophyll to sheep’s cud chewing transformed to the wiry gray mat of wool dyed pink, now holding aloft 26 spent juice containers wobbling in the wind the whole of our English alphabet.

———————————-

Midnight Recital

Kneeling to untangle my dog’s leg from its leash,
how did I get here, walking a pit bull in the dark
under the sour leaves of drought resistant Texas oaks?
How have these years colluded to put me
with a woman who doesn’t like to be touched
as if my life were still attached
to a former life, lived in felt robes, kneeling,
Why do I sometimes whisper beatitudes in Latin
when grinding roasted coffee beans for breakfast?
Why can’t a fuck be just a fuck like breathing
or the necessary forward movement of starlight
entering my eyes from Polaris when I look up?

Why is my life so intertwined that it folds me
into fractal compartments that expand, as if
from each decision, outward, new enclosures grip me
as I venture forward, faster than any logic I can conjure?
Should I kill politicians to address society’s wrongs?
Or open a shop and sell cracked imported Chinese
Chia Pets? Or get to the lunar surface to erase
the names of loved ones astronauts left behind?
How can this sticky motion of salt and water
hoisted on these dry branches of bone
discern a purpose, lost among thin pricks of starlight
that amble like ancient animals into the night?

———————————————————

Put Some Relish on Your Plate, Pontius Pilate

I started out believing in everything,
the open field, plow in hand, horse
waiting to be worked, words
hedged in the furrow, irises
open to the moment of opening

as if posturing a proof were proof enough
but without the heavy lifting of burdens,
the concrete blunders one must make, clearing
the way to ubiquitous insight.
If only my own desires would stop

helping the scrunched imp of all these days
rolled-up into aphasias of dreaming
that stream down like drops of sunlight
through the wet branches of Spring
it might be enough. Perhaps

and you will tell me everything I have every wanted
was within reach
if only I would have put out
my hands, wide palms like bells ringing

as they clap at a wedding, a wake or just praise
at the hours and minutes granted to us
I don’t know.

Put your fears in a little box and smoke it
not like this warm interrogatory weather
we’ve been having, that peels
shirts from bodies with an utter unconcern that is neither
here nor there.

———————————-

Everest

I grasp the impulse that might be driving you
to pity me in some odd way for being flabby and fifty
to your skinny and twenty, but you know, I like most
people stopped aging in my head at twenty-one, the
mental self-image of a nonstop Sid vicious, smiling at
you still trying to figure yourselves out, while we
older folk are done with nothing and wondering
everywhere we still can, asking better questions than
the thin shit we dredged up in our well-spent
grassy laid bare-assed whistling halleluiah youth.
And you listen to nothing we say all day with piercing
eyes as we watch you climbing our mistakes.

Graph Isomorpshism and Coloring

This week, Professor Siemion Fajtlowicz assigned two home work problems:

1. How many graphs with vertices 1 … n are there?

2. Up to isomorphism, how many graphs are there with n vertices?

3. If you invite 6 random people to a party, show that 3 of them will know each other or 3 of them will be mutual strangers — and show that this is guaranteed to always be the case — but only if you invite a minimum of 6 random people to the party. It will not be the case if we only invite 5 random people or 4 random people, et cetera.

____________________________________________________________________________________________________________________________
Question 1 and 2 may appear to be identical questions, but they are not at all identical though they are related.

Question 1 is asking for the number of elements of S. Question 2 is asking for the number of elements of the quotient G\S, a very different and much more difficult question.

Theorem. There are

$\displaystyle {{2}^{\left( \begin{matrix} n \\ 2 \\ \end{matrix} \right)}}$

graphs with vertex set $\displaystyle \{1,2,3,...,n-1,n\}$ .

Proof. The question is easily answered. Given n vertexes, if we start from vertex 1 and connect an edge to each of the remaining vertexes, so that 1 goes to 2, 1 goes to 3, and so on until 1 goes to (n-1) , and 1 goes finally to n. Now, including the null edge or simply 1 itself connected to nothing, which is a legitimate graph, we have that there are n graphs. Repeating this for each additional vertex, and taking into account that the edge may originate at either one of the vertexes making for two graphs — if allowing full duplicates — then we will have that there are exactly,

$\displaystyle ^{1+2+3+...+(n-1)+n}$

possible graphs raised to the power of 2 to account for starting at either vertex, which gives us:

$\displaystyle {{2}^{1+2+3+...+(n-1)+n}}={{2}^{\sum\limits_{i/{{\mathbb{Z}}_{+}}}{i}}}={{2}^{\frac{n(n+1)}{2}}}$

Therfore, the number of graphs with vertices 1 to n are:

$\displaystyle {{2}^{\left( \begin{matrix} n \\ 2 \\ \end{matrix} \right)}}$

and this is simply a more compact way of writing the RHS of the previous result using binomial coefficients. QED

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Theorem. Up to isomorphism, there are

$\displaystyle \frac{1}{n!}\sum\limits_{n\in \sum\nolimits_{n}{{}}}{{{2}^{o\left( \sigma \right)}}}$

graphs with n vertices.

The answer to the question is much more difficult, because the answer involves some hefty but basic machinery from Group Theory and Combinatorics involving Burnside’s Lemma and Polya’s Enumeration Theorem such that this question may be reworded in those terms, so that it becomes:

How large is the set $\displaystyle \sum\nolimits_{n}{{}}\backslash S$ where $\displaystyle \sum\nolimits_{n}{{}}$ denotes the symmetric group on n letters and $\displaystyle S$ is the set of graphs with vertex set $\displaystyle {1,2,3,...,n-1,n}$ ?

$\displaystyle \frac{1}{n!}\sum\limits_{n\in \sum\nolimits_{n}{{}}}{{{2}^{o\left( \sigma \right)}}}$

where $\displaystyle o\left( \sigma \right)$ denotes the number of $\displaystyle \sigma$ orbits on the set.

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At any party with 6 guests, either 3 are mutual friends or else 3 are mutual strangers. That is, the symmetric Ramsey Number R(3,3) = 6

If we consider just one person, then the other five must fall into one of the two classes of being either a friend or a stranger. This follows immediately from the pigeonhole principle, namely that, if m pigeons occupy n holes where $\displaystyle n , then at least one hole contains:

$\displaystyle \left\lfloor \frac{m-1}{n} \right\rfloor +1$

pigeons, where $\displaystyle \left\lfloor {} \right\rfloor$ is the greatest integer function. The proof of this follows from the fact that the largest multiple of n that divides into m is the fractional part left over after n divides m-1 or

$\displaystyle \left\lfloor \frac{m-1}{n} \right\rfloor$

and so for n pigeons, we get

$\displaystyle n\left\lfloor \frac{m-1}{n} \right\rfloor$

pigeons that could be put into $\displaystyle \left\lfloor \frac{m-1}{n} \right\rfloor$ holes. But we have m pigeons, and so there must be more than this many pigeons in the holes.

Now, for our problem, we have two classes, friends and strangers. If we choose one person, then that leaves 5 people with whom that person is either a friend or a stranger. And so, by the pigeonhole principle, for five objects going into two classes we must have at least:

$\displaystyle \left\lfloor \frac{5-1}{2} \right\rfloor +1=3$

members in one of the classes. In terms of a graph theoretical viewpoint, we have, starting with one vertex extended out to the other five “people” vertexes, that:

where the red edges represent friends and the blue edges represent strangers. We easily see that, regardless of how we choose one of our 2 colors, 3 of them must be blue or else 3 of them must be red, for if we have five red, then we have met the required condition that at least 3 be red, and likewise if 4 are red, we again fulfill the requirement that at least 3 be red; and so, this whole sequence of argument applies if we swap blur for red. Therefore, we are always guaranteed that there are 3 mutual friends or else that there are 3 mutual strangers in any group of 6 people. QED

In terms of Ramsey Numbers, the statement would be written $\displaystyle R(3,3)=6$ .

This is utterly fascinating, because what this is really telling us it that there is a type of structure built into any random finite set. In this case, for any binary operation or else any question or property that has two values, we have it that any finite set of 6 is sufficient to support there being 3 of one or 3 of the other of that property, and that one of the two sets of 3 always exists inside of the 6 items.

This is a glimpse into a type of “deep structure” embedded within the fabric of finite sets. This is more than merely surprising, as one should not really be expecting to find any such structure whatsoever in a random set.
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Going somewhat beyond this homework problem, from experimental math studies using Mathematica, I conjecture that:

$\displaystyle R(n,n)=\frac{1}{12}(3{{n}^{4}}-20{{n}^{3}}+63{{n}^{2}}-82n+48)$

holds for n<=20 in a completely trivial and weak algebraic sense that the Ramsey Numbers in this range do indeed appear in this formula.

However, there is no natural or reasonable theoretical connection whatsoever between this formula and Ramsey Numbers other than the search for generating functions and sequence matching studies that I conducted. So, for n = 3 to 20 this formula yields:

$\displaystyle \text{R(3,3) = 6}$
$\displaystyle \text{R(4,4) = 18}$
$\displaystyle \text{R(5,5) = }49$
$\displaystyle \text{R(6,6) = }116$
$\displaystyle \text{R(7,7) = }242$
$\displaystyle \text{R(8,8) = }456$
$\displaystyle \text{R(9,9) = }793$
$\displaystyle \text{R(10,10) = }1,294$
$\displaystyle \text{R(11,11) = }2,006$
$\displaystyle \text{R(12,12) = }2,982$
$\displaystyle \text{R(13,13) = }4,281$
$\displaystyle \text{R(14,14) = }5,968$
$\displaystyle \text{R(15,15) = }8,114$
$\displaystyle \text{R(16,16) = }10,796$
$\displaystyle \text{R(17,17) = }14,097$
$\displaystyle \text{R(18,18) = }18,106$
$\displaystyle \text{R(19,19) = }22,918$
$\displaystyle \text{R(20,20) = }28,634$

which all fall nicely into current best-known intervals for these numbers. But these values are essentially nonsense. However, it is currently believed that $\displaystyle \text{R(5,5) = }43$ rather than 49. But there is no polynomial expression that yields 43 and the other lower already known numbers in the sequence of Ramsey Numbers evident from running long and large searches for such. Therefore, I believe that any meaningful approximate or asymptotic formula must be non-polynomial.

Therefore, my intuition says that there may well exist an exponential, non-polynomial expression for the Ramsey Numbers — perhaps similar to the P(n) function of Rademacher such as

$\displaystyle p(n)=\frac{1}{\pi \sqrt{2}}\sum\limits_{k=1}^{\infty }{\sqrt{k}}{{A}_{k}}(n)\frac{d}{dn}\left( \frac{1}{\sqrt{n-\frac{1}{24}}}\sinh \left[ \frac{\pi }{k}\sqrt{\frac{2}{3}\left( n-\frac{1}{24} \right)} \right] \right)$

where

$\displaystyle {{A}_{k}}(n)=\sum\limits_{0\le m

But then I may be dreaming here because there need be no sequential connection between one of these numbers and the other — if each is a unique value in it’s own problem space.

What I find frustrating with papers on Ramsey Numbers that I have read is their lack of a more probing approach. We know that calculating Ramsey Numbers is NP-hard. One paper even suggested that this is Hyper-NP hard — but did not specify in what manner they meant this to be true. Most likely they were referring to the absurdly rapid exponential growth of the possible solution space. But where are the more basic insights into the nature of these numbers? Most of what we have now is not much beyond Paul Erdős work in the 1930’s!

In a recent paper on Ramsey Numbers, the physicist Kunjun Song, said: “Roughly speaking, Ramsey theory is the precise mathematical formulation of the statement: Complete disorder is impossible. or Every large enough structure will inevitably contain some regular substructures. The Ramsey number measures how large on earth does the structure need to be so that the speci ed (sic.) substructures are guaranteed to emerge.”

I think that we have yet to ask ourselves the deeper questions to get further along here. I am now researching the different ways in which the same questions may be asked, such as via Shannon limits of graphs and quantum algorithms to see what pure mathematical insight might be gleamed from these approaches. I am looking for good questions that, if properly phrased, should provide a road-map for further fruitful research.