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Results tagged with co.combinatorics
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user 1946
Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.
14
votes
1
answer
560
views
Is there an elementary proof of a better result for the finite guessing-box puzzle?
The infinitary guessing-box puzzle is amazing — see here. In the basic form, the Guessing-box Hall has infinitely many wooden boxes, each containing a real number, and there are 100 mathematicians wh …
14
votes
Size of maximal intersecting families
The answer is yes.
Consider first for simplicity the case where $X$ is countably infinite. If $\mathcal{S}$ is a maximal intersecting family, then I claim that $\mathcal{S}$ must contain a set with in …
6
votes
Accepted
Is every finite poset a subset of a finite complemented distributive lattice?
As Sam mentioned in the comments, the answer to question 1 is yes. One can map every condition $p$ in the partial order to the lower cone, the set $S_p=\{q\in P\mid q\leq p\}$ of conditions below $p$. …
2
votes
Accepted
Can $\omega$ be parity-separated with finitely many bijections?
No, because if you have $n$ functions, then the number of possible parity patterns to be exhibited by a number with respect to them is $2^n$. So by the pigeon-hole principle there must be infinitely m …
8
votes
Seymour's second neighborhood conjecture for infinite graphs
Allow me to make an observation concerning what I find to be an interesting angle on the question in the context without the axiom of choice, where there are competing conceptions of what it means to …
9
votes
Sunflowers in maximal almost disjoint families
It is consistent with ZFC that the answer to your question is no. Specifically, I claim, if we assume the continuum hypothesis, then the answer is no, not even for sunflowers of size $n=3$.
Theorem. A …
3
votes
Combinatorially defined effectively closed set
The answer is yes.
Let me first describe a general method. Fix any c.e. computably inseparable pair $A$ and $B$. These are computably enumerable sets having no computable separation. There are diver …
3
votes
Accepted
Is following function a metric on the set of isomorphism classes of graphs with countably ma...
To prove that this is a metric, consider the following theorem.
Theorem. If the second player can survive for $n$ steps in the $(\Gamma_1,\Gamma_2)$ game, and for $m$ steps in the $(\Gamma_2,\Gamma_3 …
2
votes
Infima and suprema in the "transfer" function ordering
Here is a counterexample showing that the quotient of $\text{Fct}(X,Y)$ is not necessarily a lattice.
Let $X=\{0,1,2\}$ and let $Y=\{0,1\}$. Let $g(0)=g(1)=0$ and $g(2)=1$, while $f(0)=0$ and $f(1)=f …
2
votes
Accepted
Can the union of difference sets in towers equal $\omega$?
Take any increasing tower, but then modify it by adding all the numbers below $n$ to $A_n$, when $n$ is even, and removing them when $n$ is odd. This is a finite change to each set in the tower, and s …
16
votes
Alice and Bob playing on a circle
For even $n$, I claim that nobody has a winning strategy, and therefore both players have drawing strategies.
To see this, observe first that by the fundamental theorem of finite games, we know that …
9
votes
3
answers
1k
views
The Sudoku game: Solver-Spoiler variation
Consider the Sudoku Solver-Spoiler game, a natural variation of the Sudoku game recently appearing in the question Who wins two-player Sudoku? posted by user PyRulez. In that game, the players attempt …
19
votes
3
answers
1k
views
The arithmetic progression game and its variations: can you find optimal play?
Consider the arithmetic progression game, a two-player game of
perfect information, in which the players take turns playing
natural numbers, or finite sets of natural numbers, all distinct,
and the fi …
58
votes
Accepted
Does knight behave like a king in his infinite odyssey?
Consider the following open knight's tour on a $5\times 5$ board, starting at position $1$ and then touring the $5\times 5$ board in the indicated move order. The final position is $25$, from which th …
134
votes
What is a chess piece mathematically?
In terms of mathematical analysis and combinatorial game theory,
the essence of any game is captured by its game tree, the tree
whose nodes represent the current game state, and to make a move in
the …