TY - JOUR

T1 - Ramsey Theory, integer partitions and a new proof of the Erdos-Szekeres Theorem

AU - Moshkovitz, Guy

AU - Shapira, Asaf

PY - 2014/9/10

Y1 - 2014/9/10

N2 - Let H be a k-uniform hypergraph whose vertices are the integers 1, . . . , N. We say that H contains a monotone path of length n if there are x1 < x2 < ⋯ < x n +k -1 so that H contains all n edges of the form {x i, x i +1, . . . , x i +k -1}. Let N k(q, n) be the smallest integer N so that every q-coloring of the edges of the complete k-uniform hypergraph on N vertices contains a monochromatic monotone path of length n. While the study of N k(q, n) for specific values of k and q goes back (implicitly) to the seminal 1935 paper of Erdos and Szekeres, the problem of bounding N k(q, n) for arbitrary k and q was studied by Fox, Pach, Sudakov and Suk.Our main contribution here is a novel approach for bounding the Ramsey-type numbers N k(q, n), based on establishing a surprisingly tight connection between them and the enumerative problem of counting high-dimensional integer partitions. Some of the concrete results we obtain using this approach are the following:•We show that for every fixed q we have N3(q,n)=2Θ(nq-1), thus resolving an open problem raised by Fox et al.•We show that for every k ≥ 3, Nk(2,n)=2{dot operator}{dot operator}2(2-o(1))n where the height of the tower is k - 2, thus resolving an open problem raised by Eliáš and Matoušek.•We give a new pigeonhole proof of the Erdos-Szekeres Theorem on cups-vs-caps, similar to Seidenberg's proof of the Erdos-Szekeres Lemma on increasing/decreasing subsequences.

AB - Let H be a k-uniform hypergraph whose vertices are the integers 1, . . . , N. We say that H contains a monotone path of length n if there are x1 < x2 < ⋯ < x n +k -1 so that H contains all n edges of the form {x i, x i +1, . . . , x i +k -1}. Let N k(q, n) be the smallest integer N so that every q-coloring of the edges of the complete k-uniform hypergraph on N vertices contains a monochromatic monotone path of length n. While the study of N k(q, n) for specific values of k and q goes back (implicitly) to the seminal 1935 paper of Erdos and Szekeres, the problem of bounding N k(q, n) for arbitrary k and q was studied by Fox, Pach, Sudakov and Suk.Our main contribution here is a novel approach for bounding the Ramsey-type numbers N k(q, n), based on establishing a surprisingly tight connection between them and the enumerative problem of counting high-dimensional integer partitions. Some of the concrete results we obtain using this approach are the following:•We show that for every fixed q we have N3(q,n)=2Θ(nq-1), thus resolving an open problem raised by Fox et al.•We show that for every k ≥ 3, Nk(2,n)=2{dot operator}{dot operator}2(2-o(1))n where the height of the tower is k - 2, thus resolving an open problem raised by Eliáš and Matoušek.•We give a new pigeonhole proof of the Erdos-Szekeres Theorem on cups-vs-caps, similar to Seidenberg's proof of the Erdos-Szekeres Lemma on increasing/decreasing subsequences.

KW - Antichains

KW - Hypergraph

KW - Integer partition

KW - Posets

KW - Ramsey Theory

UR - http://www.scopus.com/inward/record.url?scp=84903388893&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2014.06.008

DO - 10.1016/j.aim.2014.06.008

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AN - SCOPUS:84903388893

SN - 0001-8708

VL - 262

SP - 1107

EP - 1129

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -