This tag is for questions asking for combinatorial structures of maximum or minimum possible size under some constraints. Typical questions ask for bounds or the exact value of the extremal size, or for the structure of extremal configurations.

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1answer
22 views

How do I find the critical values to find the maximum of this function?

The total daily profit in dollars realized by the TKK Corporation in the manufacture and sale of x dozen recordable DVDs is given by the total profit function below. $$P(x) = −0.000001x^3 + 0.001x^2 + ...
1
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0answers
20 views

resilience of graphs question

The following is a definition of the resilience of a graph w.r.t to a property $\mathcal{P}$ (Local resilience) A property $\mathcal{P}$ is said to be monotone if the property is preserved under ...
8
votes
1answer
129 views

Tricky (extremal?) combinatorics problem

Apologies for being unsure the best way to express this problem. I have 9 tables with 4 students at each table. I want to re-seat all students so no two students who have sat together ever sit ...
1
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1answer
29 views

maximum size of a $k$-intersecting antichain of $[n]$

What is the maximum size of an antichain of $[n]:=\{1,2,3,\dots,n\}$ (say $\mathcal{A}$) such that $\mid A\cap B\mid \ge k$ where $A,B\in \mathcal{A}$ and $1\le k\le n-1$? By antichain, I mean ...
1
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0answers
37 views

maximum size of a $k$-intersecting family of $[n]$

What is the maximum size of a family of subsets of $[n]:=\{1,2,3,\dots,n\}$ say $\mathcal{A}$ such that $\mid A\cap B\mid \ge k$ where $A,B\in \mathcal{A}$ and $1\le k\le n-1$? This not ...
1
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0answers
17 views

Szemeredi Trotter and additive combinatorics on A+AA

I am trying to get a lower bound on $|A+AA|$ where $A$ is a set, and $A+AA=\{a+bc: a,b,c \in A\}$ using Szemeredi Trotter. I would think we need to form lines of the form $y=ax+b$ where $a,b \in A$, ...
4
votes
1answer
146 views

Application of Erdős–Ko–Rado

Here is an interesting question. I believe you have to use the fact the Erdős–Ko–Rado Theorem tells you $A$ and $B$ are not intersecting, but I am unable to show it: Let $A,B \subset[n]^{(r)}$, ...
2
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1answer
71 views

Generalization of Erdős-Ko-Rado where intersections have cardinality in given set

Problem: Let $T\subseteq \{0,\ldots, k-1\}$ and let $\mathcal F\subseteq [n]^{(k)}$ (subsets of size $k$) such that $|A\cap B|\in T$ for $A,B\in \mathcal F$, $A\ne B$. Show that if $n\ge ...
10
votes
1answer
114 views

Which graphs can be drawn using straight lines with no disjoint edges?

What is the class of graphs that can be drawn using only straight lines with no two edges disjoint? Edges are disjoint when they don't cross and they don't share a vertex. Vertices should be in ...
3
votes
1answer
82 views

Number of edges in graphs having two disjoint cycles of equal length

The question is motivated by this and this two problems. The first problem states that if $G$ is a graph with $n$ vertices and at least $2n-2$ edges then $G$ must contain two distinct cycles of the ...
2
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3answers
144 views

Pigeonhole Principle to Prove a Hamiltonian Graph

I am trying to figure out if a graph can be assumed Hamiltonian or not, or if it's indeterminable with minimal information: A graph has 17 vertices and 129 edges. ...
2
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0answers
30 views

Covering the square with “crosses”.

The problem concerns covering the unit square with translates of a specific figure, which I will refer to as a "cross", using as few translates as possible. The difficulty seems to result from the ...
2
votes
1answer
34 views

Maximum independent sets of balanced bipartite graph

Suppose that $G=(V,E)$ is a connected bipartite graph with $|V|=N$ and vertex set bipartition $V = A \cup B$ such that $|A|=|B|$. Assume that $\alpha(G) = N/2$. Is it always true that $A$ and $B$ are ...
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0answers
54 views

zarankiewicz problem lower bound

I was just reading through the following article: http://page.mi.fu-berlin.de/szabo/PDF/stoc96.pdf On page 2 they give an explicit formula for the lower bound of the size of the graph. Summary: We ...
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0answers
13 views

Want to construct a skew starter for the skew room square of order 667.

If I can obtain a skew starter for the skew room square of order 667 then I can construct the symmetric block design (667,333,166). Then once I have the incidence matrix M for this design, I can then ...
-2
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1answer
42 views

Min $A=14(a^{2}+b^{2}+c^{2})+\frac{ab+bc+ac}{a^{2}b+b^{2}c+c^{2}a}$?

$a;b;c\in \mathbb{R}^+$ such that $a+b+c=1$. Find the minimum of $A=14(a^{2}+b^{2}+c^{2})+\frac{ab+bc+ac}{a^{2}b+b^{2}c+c^{2}a}$
2
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0answers
28 views

Threshold function of the property Sperner set

Someone can to help me in the following problem about the threshold function of the property Sperner set? I don't know where to start. Let $\mathcal{F} \subseteq \mathcal{P}([n])$ be a random ...
5
votes
3answers
383 views

Show that if there are 101 people of different heights standing in a line

Show that if there are 101 people of different heights standing in a line, it is possible to find 11 people in the order they are standing in the line with heights that are either increasing or ...
2
votes
1answer
111 views

Linear algebra and combinatorics. For a family with even size sets and even intersections prove that $|F| \le 2^{n/2}$

Let $F \subset P(n)$ be a family such that for all i and j $ |f_i \cap f_j|$ and $|f_i|$ are even Prove that $|F| \le 2^{n/2}$ Now I think we go by contradiction and say if $|F| \ge 2^{n/2}+1$ ...
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0answers
34 views

Kruskal-Katona Theorem with Majority?

I am interested in the following problem which seems like an extension of the Kruskal-Katona Theorem. Let $A_k \subseteq \{0,1\}^n$ be a subset of the hypercube such that every element in $A$ has ...
0
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2answers
75 views

Proved : $3(a+b+c)\geq \sqrt{8a^{2}+1}+\sqrt{8b^2+1}+\sqrt{8c^{2}+1}$

$1$. $x;y;z\in \mathbb{R}$ such that $xyz=1$. Find the minimum or maximum value of : $\sum \dfrac{1}{x+1}$ $2$. $x;y;z\in \mathbb{R}^+$ such that $a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$. Proved ...
2
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1answer
35 views

What is the first $w$ such that a rectangle, $R_{w\times w-1}$ is minimally-square-partitioned by less than $w$ squares.

Motivated by: Tiling an orthogonal polygon with squares, How to prove that the minimum square partition of a 3X2 rectangle has 3 squares, Minimum square partitions for 4x3 and 5x4 rectangles, What is ...
0
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1answer
41 views

If $|V(G)|=n$ and $e(G)>\frac{n}{4}\{1+\sqrt{4n-3}\}$ then $G$ contains 4-cycle

This question is linked to my former question Special properties of subgraphs I want to practice this technique a little bit more and want to show that if $|V(G)|=n$ and ...
2
votes
1answer
165 views

Minimum square partitions for 4x3 and 5x4 rectangles

Motivation: Tiling an orthogonal polygon with squares Followup question: What is the first $w$ such that a rectangle, $R_{w\times w−1}$ is minimally-square-partitioned by less than $w$ squares.. Yes, ...
0
votes
1answer
61 views

Convergence of $\text{ex} (n;P)/ \binom n2$ for Petersen graph

This question is linked to For a graph $G$, why should one expect the ratio $\text{ex} (n;G)/ \binom n2$ to converge? where an argument was given that this specific ratio converges for ...
3
votes
1answer
54 views

Minimum boolean lattice containing all poset of fixed size

I need help with the following: What is the minimum $n$ such that the boolean lattice $2^{[n]}$ contains all posets of size $m$? I noticed that it should contain a chain of length $m$, and the ...
2
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1answer
129 views

LYM Inequality question

Suppose that $F ⊂ P(n)$ is a set system containing no chain with $k + 1$ sets. Prove that $\sum\limits_{r=1}^n \frac{|F_{r}|}{n \choose r} ≤ k$, where $F_{i} = F \cap [n]^{(i)}$ for each i. ...
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2answers
80 views

Let $k \le \frac{n}{2}$, and suppose that $F$ is an antichain in $P(n)$ such that every $A \in F$ has $|A| \le k$. Prove that $|F| \le \binom{n}{k}$

I'm stuck on this combinatorics question: Let $k \le \frac{n}{2}$, and suppose that $F$ is an antichain in $P(n)$ such that every $A \in F$ has $|A| \le k$. Prove that $|F| \le \binom{n}{k}$. I've ...
0
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1answer
42 views

Shadow of a set system

I'm currently learning something about Sperner's Lemma and then the LYM Inequality. In trying to prove the LYM Inequality, the proof uses the concept of a shadow but I can't seem to get a proper grip ...
2
votes
1answer
58 views

Maximal rectangle in a permutation

Suppose you have a permutation of n elements, and it is represented by colouring squares in a n by n grid of squares, where only one square is coloured in each row or column. Find the minimum area of ...
1
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1answer
37 views

Balanced independent sets & independent domination number

Let $G=(V,E)$ be a bipartite graph, with partition $V=A \cup B$. Recall that an independent set $I$ of $G$ is a set of vertices sharing no edges. The independent domination number $i(G)$ is defined ...
6
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1answer
257 views

Minimum number of lines covering n points

Let there be n points in the plane. I want to know the minimum number of horizontal and vertical lines covering all the points in the plane. My initial approach started like this, 1) for each point I ...
2
votes
1answer
34 views

Error correcting binary partition

Let's say I have a collection of $2^n$ labeled objects, and I want to find one of them. If I can ask yes-no questions about it, binary partition would immediatly lead us to the desired object in $n$ ...
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0answers
62 views

Intersecting Set Systems

This a follow up question to an earlier question: "Intersecting set systems and Erdos-Ko-Rado Theorem " that was answered by Sean Eberhad. It asks, what is the largest intersecting family of r-sets ...
5
votes
1answer
123 views

Number of combinations such that each pair of combinations has at most x elements in common?

I am doing research on the sense of smell and have a combinatorics problem: I have 128 different odors (n) and I mix them in mixtures of 10 (r). There are 2.26846154e+14 different mixtures. What I ...
9
votes
3answers
490 views

How many non-isomorphic ways a convex polygon with $n + 2$ sides can be cut into triangles?

From Wikipedia: The Catalan number $C_n$ is the number of different ways a convex polygon with $n + 2$ sides can be cut into triangles by connecting vertices with straight lines (a form of Polygon ...
3
votes
1answer
113 views

Suppose there are k points, no 3 of which are collinear. What is the upper bound on the number of quadrilaterals we can form?

We understand that the number of triangles possible is given by kC3, since any selection of 3 points uniquely determine a triangle. This is not true for quadrilaterals though, since for selections ...
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1answer
58 views

Has this variation of blocking set been studied before?

Given a collection of sets $F$, a set which intersects all sets in the $F$ in at least one element is called a blocking set (or hitting set). The blocking number $τ(F)$ of a family $F$ is the minimum ...
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0answers
48 views

An extremal combinatorial problem over Finite rings

Let $q$ be an odd number and $g_i = (g_{i1} g_{i2} \dots g_{ir}) \in \Bbb Z_q^r$ a list of vectors with $i\in\{1,\ldots,L\}$. Let each $g_i$ have $0 < k < r$ zero entries. What is the maximum ...
5
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1answer
133 views

Show there’s at most $n\choose \left \lfloor\frac{n}{2} \right\rfloor$ subsets $A\subset[n]$ such that $\displaystyle\sum\limits_{i\in{A}} a_i=\alpha$

Let $a_1, a_2, a_3, ... , a_n$ and $\alpha$ be n+1 non-zero real numbers. Prove that there are at most $n\choose \left\lfloor\frac{n}{2}\right\rfloor$ subsets $A\subset[n]$ such that ...
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2answers
124 views

Counting the number of subgraphs isomorphic with the following digraph

Suppose $H$ is the following 4-vertex digraph : $$\langle V=\{a,b,c,d\}, E=\{ab,bd,ac,cd\}\rangle .$$ The digraph is drawn below: Can one help me to prove upper bound $n^4/55$ on the number of ...
1
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1answer
728 views

Prove that the maximum number of edges in a graph with no even cycles is floor(3(n-1)/2)

The question is in the title. I can see why the bound is sharp (for example, a lot of triangles sharing one common vertex if n is odd, or the same but with one spare edge hanging out if n is even). ...
6
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0answers
93 views

Biggest Little Polyhedron

The Biggest Little Polygon problem asks for the polygon with greatest area where the largest diameter is 1. Let's add a dimension and find the largest volumes. What is the biggest little polyhedron ...
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0answers
49 views

Difference Sets

suppose we have a set $$P=\{p_1,p_2,...,p_K\}$$ where $$1\leq p_k\leq N , k=1,...,K \qquad \& \quad p_k \in \mathbb{N} $$ and $p_k$'s are distinct. We calculate the differences as: $$d=p_i-p_j\mod ...
5
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0answers
76 views

Is Paley-13 graph a unit distance graph in 3D space?

The 13-node Paley graph has vertices 1 to 13 that are connected by an edge when their difference is one of the values $(1,3,4,9,10,12)$ Can this graph be put into 3D space so that all edges have ...
2
votes
1answer
234 views

Maximum number of pairwise intersections

Let $[n]=\{1,2,\ldots,n\}$ and let $S$ consist of subsets of $[n]$ of cardinality $2$. I would like to find the maximum number of pairwise intersections that $k$ distinct elements from $S$ can have. ...
1
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1answer
57 views

beads in a string with restrictions

Using the principle of inclusion and exclusion; let $p,q\in \mathbb{N}$, being $p$ odd, there are $pq$ beads of $q$ different colors, with $p$ beads in each color. If the beads of the same colour are ...
2
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1answer
150 views

minimum lines, maximum points

There are $P$ points in the 2-dimensional plane. Through each point, we draw two orthogonal lines: one horizontal (parallel to x axis), one vertical (parallel to y axis). Obviously, some of these ...
1
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1answer
25 views

Find maximum size of set family, where at least one member has at least one of any three points, but do not contains all those three points

Let $n$ and $s$ be non-negative integers such that $n\geq 3$ and $2s\log2>3\log n$. Prove that there exists sets $A_1,A_2,\ldots,A_s \subseteq [n]$ such that for every $B\in \binom {[n]} 3$ there ...
1
vote
1answer
59 views

Maximum size of k-uniform set family that satisfies a condition

Let $n \leq 2k$ and $A_1,A_2,...,A_m \subseteq [n]$ be distinct $k$-uniform set where $A_i\cup A_j \neq [n]$ for all $1 \leq i < j \leq m $. Prove that $$m \leq (1- \frac{k}{n}) \binom n k$$ ...