An old math puzzle goes: Suppose there are six people in a room; some of them shake hands. Prove that there are at least three people who all shook each others’ hands or three people such that no pair of them shook hands. Generalized a bit, this is the classic Ramsey problem. The diagonal Ramsey numbers R(k) are de ned as follows. R(k) is the smallest integer n such that in every two-coloring o...

The typical problem in (generalized) Ramsey theory is to find the order of the largest monochromatic member of a family F (for example matchings, paths, cycles, connected subgraphs) that must be present in any edge coloring of a complete graph Kn with t colors. Another area is to find the minimum number of monochromatic members of F that partition or cover the vertex set of every edge colored c...

Let Kk,n be a complete k-partite graph of order n and let K k,n be a generalized complete k-partite graph of order n spanned by the fan set F = {Fn1 , Fn2 , · · · , Fnk}, where n = {n1, n2, · · · , nk} and n = n1 + n2 + · · ·+ nk for 1 6 k 6 n. In this paper, we get the number of spanning trees in Kk,n to be t(Kk,n) = n k−2 k ∏ i=1 (n− ni)i. and the number of spanning trees in K k,n to be t(K k...

Let the random variable Zn,k denote the number of increasing subsequences of length k in a random permutation from Sn, the symmetric group of permutations of {1, ..., n}. We show that V ar(Zn,kn ) = o((EZn,kn ) ) as n → ∞ if and only if kn = o(n 2 5 ). In particular then, the weak law of large numbers holds for Zn,kn if kn = o(n 2 5 ); that is, lim n→∞ Zn,kn EZn,kn = 1, in probability. We also ...

Higher-order Fourier analysis, developed over prime fields, has been recently used in different areas of computer science, including list decoding, algorithmic decomposition and testing. We extend the tools of higher-order Fourier analysis to analyze functions over general fields. Using these new tools, we revisit the results in the above areas. (i) For any fixed finite field K, we show that th...

A field K is called an n-dimensional local field if there is a sequence of fields kn, . . . , k0 satisfying the following conditions: k0 is a finite field, ki is a complete discrete valuation field with residue field ki−1 for i = 1, . . . , n, and kn = K . In [9] we defined a canonical homomorphism from the n th Milnor group Kn(K) (cf. [14]) of an n-dimensional local field K to the Galois group...