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Clearly there is at most one increasing path, so the only problem is to find it. Take some path, and suppose it is not increasing. So there is some length 2 subpath which is not increasing. Repla …

If you replace S by the lattice points in the positive orthant, and forget the condition that the hyperplane passes through a particular point, and require that the hyperplane only cuts off finitely m …

The answer is no. Here is a list of sets $A_i$ and $B_i$ which fails. 12, 1234 23, 1235 13, 1236 14, 1245 25, 2356 36, 1346 45, 1456 56, 2456 46, 3456 The failure can be seen by drawing the pictu …

O(1) is impossible even if you drop the condition of no monochromatic rectangles, and even if you know that the two cells are always chosen within a given row. Suppose the length of the rows, $m$, …

There is the obvious lower bound of nN - {n \choose 2}N_2. (I'm taking N_2 to be the bound on the size of an intersection of 2 sets; I'm not sure if that's what you meant.) I don't think it's poss …

A better solution than my previous one is max_{1\leq i \leq n} iN - {i \choose 2}N_2 (That is to say, we can simply consider only i of the sets instead of all n of them, and then apply my previous …

Join semi-distributive lattices don't have this property because weak order on $S_n$ is join semi-distributive and doesn't have this property. (Eg, for $n=3$.) Lattices satisfying the property you a …

Proposition 4 of Morelli's paper "Pick's Theorem and the Todd class of a toric variety" gives a sufficient condition: it describes a setting in which there is a positive formula for coefficient of $x^ …

I think the answer is "no". Let's consider $p=2,d=3$. Suppose that we have a necklace which can be fairly divided using only 2 cuts (one less than the maximum number that may be required). Let …

This is an answer to Alexander's combinatorial reformulation of the question in comments to Bruce's answer. dim $V_\lambda$/$n$! is the chance that you will get a standard Young tableau if you assi …

This is "the same" as the generating function proof, but it doesn't use generating functions explicitly. Take the largest common difference in any of the sequences, say n, and pick $\zeta$ a primitiv …

Are there any purely combinatorial criteria which allow you to deduce that a spherical simplicial complex is polytopal (i.e., there exists a simplicial polytope whose boundary is isomorphic to it)? F …

Define a ground set $X$ of size $2^{n-1}$. Now choose $2^{n-1}-(n-1)$ subsets of $X$, each of size at least 2, such that the sum of their sizes is $(n-2)2^{n-1}+2$ (so the average size is slightly mo …

It follows from the Knuth-Netto formula that the asymptotics of $I_n(k)$, for $k$ fixed, is $n^k/k!$ to first order. I claim the asymptotic behaviour of $I_n^{\sigma(y)=x}(k)$ is as $n^{k-|x-y|}/ …

It seems unlikely (once you assume d.c.c.). Define the height of an element $x$ in $P$ to be the length of the shortest unrefinable chain from $x$ to $0$. Let $P_n$ denote the elements of $P$ whos …