In 1926 van der Waerden [13] proved the following startling theorem : If the set of integers is arbitrarily partitioned into two classes then at least one class contains arbitrarily long arithmetic progressions. It is well known and obvious that neither class must contain an infinite arithmetic progression. In fact, it is easy to see that for any sequence an there is another sequence bn9 with bn > an9 which contains no arithmetic progression of three terms, but which intersects every infinite arithmetic progression. The finite form of van der Waerden's theorem goes as follows: For each positive integer n9 there exists a least integer f{n) with the property that if the integers from 1 to /(/?) are arbitrarily partitioned into two classes, then at least one class contains an arithmetic progression of « terms. (For a short proof, see the note of Graham and Rothschild [5].) However, the best upper bound on f{n) known at present is extremely poor. The best lower bound known, due to Berlekamp [3], asserts that/(«) < nl9 for n prime, which improves previous results of Erdös, Rado and W. Schmidt. More than 40 years ago, Erdös and Turân [4] considered the quantity rk{n)9 defined to be the greatest integer / for which there is a sequence of integers 0 < a\\ < a2 < ••• < a; ^ n which does not contain an arithmetic progression of k terms. They were led to the investigation of rk{n) by several things. First of all the problem of estimating rk{n) is clearly interesting in itself. Secondly, rk{n) < n/2 would imply f{k) < 77, i.e., they hoped to improve the poor upper bound on f{k) by investigating rk{n). Finally, an old question in number theory asks if there are arbitrarily long arithmetic progressions of prime numbers. From rk{n) < %{rì) this would follow immediately. The hope was that this problem on primes could be attacked not by

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On sets of integers containing k elements in arithmetic progression

A new proof of szemer ´ edi's theorem

Old and new problems and results in combinatorial number theory

Recent advances in high-throughput experimental methods for the identification of protein interactions have resulted in a large amount of diverse data that are somewhat incomplete and contradictory. As valuable as they are, such experimental approaches studying protein interactomes have certain limitations that can be complemented by the computational methods for predicting protein interactions. In this review we describe different approaches to predict protein interaction partners as well as highlight recent achievements in the prediction of specific domains mediating protein-protein interactions. We discuss the applicability of computational methods to different types of prediction problems and point out limitations common to all of them.

/pdf/deciphering-protein-protein-interactions-part-ii-3ahhxzka1q.pdf

Deciphering protein-protein interactions. Part II. Computational methods to predict protein and domain interaction partners.

http://www.marcottelab.org/paper-pdfs/cosb-review.pdf

Protein interaction networks from yeast to human.

/pdf/euclidean-ramsey-theorems-i-3vrdeddr4n.pdf

Euclidean Ramsey Theorems. I

Ramsey theory (2nd ed.)

Finding the interacting pairs of proteins between two different protein families whose members are known to interact is an important problem in molecular biology. We developed and tested an algorithm that finds optimal matches between two families of proteins by comparing their distance matrices. A distance matrix provides a measure of the sequence similarity of proteins within a family. Since the protein sets of interest may have dozens of proteins each, the use of an efficient approximate solution is necessary. Therefore the approach we have developed consists of a Metropolis Monte Carlo optimization algorithm which explores the search space of possible matches between two distance matrices. We demonstrate that by using this algorithm we are able to accurately match chemokines and chemokine-receptors as well as the tgfbeta family of ligands and their receptors.

/pdf/inferring-protein-interactions-from-phylogenetic-distance-ho1y8xhslx.pdf

Inferring protein interactions from phylogenetic distance matrices.

Marshall Hall, Jr.

A short proof is given for the classical theorem of van der Waerden which asserts that for any partition of the integers into a finite number of classes, some class contains arbitrarily long arithmetic progressions Let [a, b] denote the set of integersxwith a_x_b We call (x1, ** ,x (x', * x*m) E [0, r]m I-equivalent if they agree up through their last occurrences of i For any 1, m> 1, consider the statement For any r, there exists N(I, m, r) so that for any function S(, m) C: [1, N(i, m, r)]-[1, r], there exist positive a, d,, * *, dm such that C(a+ Im 1 xidi) is constant on each 1-equivalence class of [0, i]m FACT 1 S(i, m)for some m >?I=S(l,m+1) PROOF For a fixed r, let M=N(I, m, r), M'=N(l, 1, rM) and suppose C: [1, MM']-[1, r] is given Define C': [1, M']-*[1, rM] so that C'(k)= C'(k') iff C(kM-j)=C(k'M-j) for all 0_ j l=>S(I+1, 1) PROOF For a fixed r, let C: [1, 2N(l, r, r)]1,r] be given Then there exist a, dl, * , dr such that for xi E [0, 1], a+ jr 1 xidj<N(i, r, r) and C(a+ 22=1 xidi) is constant on I-equivalence classes By the box principle there exist u<v in [0, r] such that C(a + E Is) =C (a+2id)Received by the editors February 1, 1973 A MS (MOS) subject classifcations (1970) Primary 05A99; Secondary lOL99

/pdf/a-short-proof-of-van-der-waerden-s-theorem-on-arithmetic-5a2j7a0zcz.pdf

Bruce Rothschild

Papers

Euclidean Ramsey Theorems. I

Ramsey theory (2nd ed.)

Inferring protein interactions from phylogenetic distance matrices.

Marshall Hall, Jr.

A short proof of van der Waerden’s theorem on arithmetic progressions