Ramsey and The Pigeonhole Principle

Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of mathematics that studies the conditions under which order must appear. Problems in Ramsey theory typically ask a question of the form: ‘how many elements of some structure must there be to guarantee that a particular property will hold?’

Illustration of The Pigeonhole Principle

Suppose, for example, that we know that n pigeons have been housed in m pigeonholes. How big must n be before we can be sure that at least one pigeonhole houses at least two pigeons? The answer is the pigeonhole principle: if n > m, then at least one pigeonhole will have at least two pigeons in it.

The photograph on the right shows a number of pigeons in holes. Here there are n = 10 pigeons in m = 9 holes, so by the pigeonhole principle, at least one hole has more than one pigeon: in this case, both of the top corner holes contain two pigeons. The principle says nothing about which holes are empty: for n = 10 pigeons in m = 9 holes, it simply says that at least one hole here will be over-full; in this case, the bottom-left hole is empty. Ramsey’s theory generalizes this principle as explained below.

A typical result in Ramsey theory starts with some mathematical structure that is then cut into pieces. How big must the original structure be in order to ensure that at least one of the pieces has a given interesting property?

For example, consider a complete graph of order n; that is, there are n vertices and each vertex is connected to every other vertex by an edge. A complete graph of order 3 is called a triangle. Now colour every edge red or blue. How large must n be in order to ensure that there is either a blue triangle or a red triangle? It turns out that the answer is 6.

Another way to express this result is as follows: at any party with at least six people, there are three people who are all either mutual acquaintances (each one knows the other two) or mutual strangers (each one does not know either of the other two).

A comic result of the pigeonhole principle is the “proof” that in the city of New York (or any other city with a population over a million) at least two people have the same number of hairs on their head. The reasoning is as follows: an average human being has about 150.000 hairs on the scalp; it is reasonable to assume that no human being has more than 1.000.000 hairs on the scalp. Over a million people live in New York. The population of n people (exceeding a million) has to be arranged in m (1.000.000 or less) collections; one collection is possible for each number of hairs on the scalp. Because n population > m different numbers of hairs on the scalp, there are at least two people in one of these collections – at least two people with the same number of hairs.

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Sesquipedality [Noun.]

Literally speaking, sesquipedality is using words that are one and a half feet long. A related word is sesquicentennial – a 150th anniversary.

Seeds of Yardlong or Yard Long Bean

A sesquipedalian word is a term for a (very) long, polysyllabic word. This dictum doesn’t apply to German speakers though, as Mark Twain once observed: “Some German words are so long that they have a perspective”.

There’s a bean subspecies commonly known as a yardlong bean. It’s really misnamed as it’s ‘only’ half a yard long. Its scientific name, Vigna unguiculata subspecies; sesquipedalis, is more precise.

Schmidt Sting Pain Index

The Schmidt Sting Pain Index is a pain scale rating the relative pain caused by different Hymenopteran stings. It is mainly the work of Justin O. Schmidt, an entomologist at the Carl Hayden Bee Research Center. Schmidt has published a number of papers on the subject and claims to have been stung by the majority of stinging Hymenoptera.

His original paper in 1984 was an attempt to systematize and compare the hemolytic properties of insect venoms. The index contained in the paper started from 0 for stings that are completely ineffective against humans, progressed through 2, a familiar pain such as a common bee or wasp sting, and finished at 4 for the most painful stings.

In the conclusion of the Pain Index, some connoisseur descriptions of the most painful examples were given, e.g.: “Paraponera clavata stings induced immediate, excruciating pain and numbness to pencil-point pressure, as well as trembling in the form of a totally uncontrollable urge to shake the affected part.”


Halitosis is a term used to describe noticeably unpleasant odours exhaled in breathing.

Scientists have long thought that smelling one’s own breath odour is often difficult due to acclimatization, although many people with bad breath are able to detect it in others. Research has suggested that self-evaluation of halitosis is not easy because of preconceived notions of how bad we think it should be. Some people assume that they have bad breath because of bad taste, however bad taste is considered a poor indicator.

For these reasons, the simplest and most effective way to know whether one has bad breath is to ask a trusted adult family member or very close confidant. If the confidant confirms that there is a breath problem, he or she can help determine whether it is coming from the mouth or the nose, and whether a particular treatment is effective or not.

One popular home method to determine the presence of bad breath is to lick the back of the wrist, let the saliva dry for a minute or two, and smell the result. This test results in overestimation, as concluded from research, and should be avoided. A better way would be to lightly scrape the posterior back of the tongue with a plastic disposable spoon and to smell the drying residue. Home tests that use a chemical reaction to test for the presence of polyamines and sulphur compounds on tongue swabs are now available, but there are few studies showing how well they actually detect the odour. Furthermore, since breath odour changes in intensity throughout the day depending on many factors, multiple testing sessions may be necessary.

A Woman’s Mouth Decorated with Lipstick

If bad breath is persistent, and all other medical and dental factors have been ruled out, specialized testing and treatment is required. Hundreds of dental offices and commercial breath clinics now claim to diagnose and treat bad breath. They often use some of several laboratory methods for diagnosis of bad breath:

Halimeter: a portable sulphide monitor used to test for levels of sulphur emissions (to be specific, hydrogen sulphide) in the mouth air. When used properly, this device can be very effective at determining levels of certain (VSC-producing) bacteria.

Gas chromatography: this technology is specifically designed to digitally measure molecular levels of the three major VSCs in a sample of mouth air (hydrogen sulphide, methyl mercaptan, and dimethyl sulphide). It is accurate in measuring the sulfur components of the breath and produces visual results in graph form via computer interface.

BANA test: this test is directed to find the salivary levels of an enzyme indicating the presence of certain halitosis-related bacteria.

β-galactosidase test: salivary levels of this enzyme were found to be correlated with oral malodour.

Although such instrumentation and examinations are widely used in clinics, the most important measurement of bad breath is the actual sniffing and scoring of the level and type of the odour carried out by trained experts. The level of odour is usually assessed on a six-point intensity scale.

Who is being Objectified?

‘This analysis of fifty of the bestselling pornographic videos in Australia shows that women are not objectified in this genre more than men . Of our twelve measures, seven can be analysed to measure gendered differentiation of objectification in pornography. We excluded the kinds of sex acts, and sex acts causing orgasm from this part of the analysis – this data is important and suggestive but cannot be compared in this particular way as there exists no agreed scale to quantify the pleasure different sex acts cause each gender. We also excluded measures of violence from gendered comparison, as they are too few in the sample to allow comparison of gender roles to be meaningful.

Of these seven measures, one shows women being more objectified than men (presence of orgasms, where women have fewer orgasms). Three show men being more objectified than women (in time spent looking at camera, where men return the gaze less; in time spent talking to the camera, where they are also less engaged; and in initiating sex, where men are more sexual objects than active sexual subjects in seeking their sexual pleasure in the sample). Three measures showed no large difference in objectification between men and women (naming, central characters and time spent talking to other characters). […]

In the mainstream of pornographic videos in Australia we found […] and a very small amount of violence – and then, only when we erred on the side of inclusiveness in deciding whether situations might be consensual or not. The majority of scenes containing violence came from videos which were explicitly marketed to women.

Overall, women were no more objectified than men in the mainstream of pornography. These results are reassuring. This is the first study, we believe, to survey and attempt to reconcile the measures employed in previous content analyses of pornography. By choosing to use the term ‘objectification’ as the key concept under which various other forms of undesirable representation (including violence) can be measured, we believe that we have offered a potentially useful new approach to the analysis of pornography; one that allows for analyses that are sensitive to the specificity of filmic representations, that work within accepted social science definition of aggression, and can be easily articulated to ongoing public debates about the genre. We hope that other researchers will take up this approach to provide a more detailed understanding of the workings of pornography across media, and across cultures.’

– McKee, Alan (2005) The Objectification of Women in Mainstream Porn Videos in
Australia. The Journal of Sex Research 42(4): p. 277-290