On the cardinality spectrum and the number of latin bitrades of order 3 By a (Latin) unitrade of order k, we call a subset of vertices of the Hamming graph H(n, k) that intersects any maximal clique at either 0 or 2 vertices. A bitrade is a bipartite unitrade, i.e., a unitrade that can be split into two independent subsets. We study the cardinality spectrum of bitrades in the Hamming graph H(n, k) with k  3 (ternary hypercube) and the growth of the number of such bitrades as n grows. In particular, we determine all possible small (up to 2.5xc2xb72n) and large (from 14xc2xb73nxe2x88x923) cardinalities of bitrades of dimension n and prove that the cardinality of a bitrade takes only values equivalent to 0 or 2n modulo 3 (this result can be treated in terms of a ternary ReedMuller type code). A part of the results are valid for an arbitrary k. For k  3 and n xe2x86x92 xe2x88x9e we prove that the number of nonequivalent bitrades is not less than 2(23xe2x88x92o(1))n and not greater than 2alphan, xcexb1 u003c 2 (the growth order of the double logarithm of this number remains unknown). Alternatively, the studied set Bn of bitrades of order 3 can be defined as follows: B0 consists of three rationals  1, 0, 1; Bn consists of ordered triples (a, b, c) of elements from Bnxe2x88x921 such that a  b  c  0.

Ordered graphs and large bicliques in intersection graphs of curves. An ordered graph G_ 0 such that every ordered graph on n vertices that does not contain a monotone path of length k as an induced subgraph has a vertex of degree at least c_kn, or its complement has a biclique of size at least c_knlog n. A similar result holds for ordered graphs containing no induced ordered subgraph isomorphic to a fixed ordered matching.  As a consequence, we give a short combinatorial proof of the following theorem of Fox and Pach. There exists a constant cu003e0 such the intersection graph G of any collection of n xmonotone curves in the plane has a biclique of size at least cnlog n or its complement contains a biclique of size at least cn. (A curve is called xmonotone if every vertical line intersects it in at most one point.) We also prove that if G has at most left(frac14 epsilonright)nchoose 2 edges for some epsilonu003e0, then overlineG contains a linear sized biclique. We show that this statement does not remain true if we replace frac14 by any larger constants.