Algebraic Methods

If every subset in $\mathscr {F}$ has an odd number of elements and the intersection of any two subsets in $\mathscr {F}$ has an even number of elements, then $\left|\mathscr {F}\right| \leq n$.

Construct an example where this bound is attained.

If the intersection of any two subsets in $\mathscr {F}$ has exactly $q$ elements and every subset in $\mathscr {F}$ has more than $q$ elements, then $\left|\mathscr {F}\right| \leq n$.

If $q > 0$ and the intersection of any two subsets in $\mathscr {F}$ has exactly $q$ elements, then $\left|\mathscr {F}\right| \leq n$.

There exist $2^{k}$ subsets of a $2k$-element set such that each of them has an even number of elements and the intersection of any two of them has an even number of elements.

There cannot be more than $2^{k}$ subsets satisfying the conditions of part (1).

The largest number of points in $\mathbb {R}^{n}$ with equal pairwise distances is $n+1$.

Construct $\dfrac {n(n-1)}{2}$ points in $\mathbb {R}^{n}$ whose pairwise distances take only two distinct values.

For $a \in \mathbb {R}$ and points $v = (v_{1}, \ldots , v_{n}) \in \mathbb {R}^{n}$, $x = (x_{1}, \ldots , x_{n}) \in \mathbb {R}^{n}$, denote $P_{v}(x) := \left|x-v\right|^{2} - a^{2}$. If the pairwise distances between $k$ points $u_{1}, \ldots , u_{k} \in \mathbb {R}^{n}$ are equal to $a$, then the polynomials $P_{u_{1}}, \ldots , P_{u_{k}}$ are linearly independent over $\mathbb {Q}$.

If the pairwise distances between $k$ points in $\mathbb {R}^{n}$ take only two distinct values, then $k \leq \dfrac {(n+1)(n+4)}{2}$.

Among any $327$ pairwise intersecting $9$-element subsets of a $25$-element set, there are two subsets whose intersection has exactly $3$ or exactly $6$ elements.

Among any $327$ points in $V_{25,9}$, there are two whose scalar product lies in $\left\{ 0,3,6\right\}$.

For any $\overrightarrow {a} \in V_{25,9}$, expand the product

where $x_{1}, x_{2}, \ldots , x_{25}$ are variables. In each of the resulting monomials, perform the following operation: for each $i$, if the monomial contains the factor $x_{i}^{2}$, replace this factor with $1$. Denote the resulting polynomial by $F_{\overrightarrow {a}}(x_{1}, \ldots , x_{25})$. Prove that if the scalar product of no two vectors among $\overrightarrow {a}_{1}, \ldots , \overrightarrow {a}_{s} \in V_{25,9}$ is divisible by $3$, then the polynomials $F_{\overrightarrow {a}_{1}}, \ldots , F_{\overrightarrow {a}_{s}}$ are linearly independent over $\mathbb {Q}$.

Specify $326$ polynomials whose linear combinations with rational coefficients yield every polynomial $F_{\overrightarrow {a}}$, $\overrightarrow {a} \in V_{25,9}$.

Among any $107$ five-element subsets of a $14$-element set, there are two subsets whose intersection has exactly $2$ elements.

The same for $93$ subsets.

The same for $92$ subsets.

It is impossible to color all five-element subsets of a $14$-element set with $21$ colors so that any two five-element subsets intersecting in exactly two elements have different colors.

For a prime $p$ and an integer $t$, the number $G(t) := (t-1)(t-2)\ldots (t-p+1)$ is divisible by $p$ if and only if $t$ is not divisible by $p$.

Let $p$ be a prime and $n = 4p$. Denote

Denote $G(t) := (t-1)(t-2)\ldots (t-p+1)$. For any $\overrightarrow {a} \in M$, expand the brackets in the product $G(\overrightarrow {a} \cdot (1, x_{2}, \ldots , x_{n}))$, where $x_{2}, \ldots , x_{n}$ are variables. In each of the resulting monomials, for each $i$ we replace $x_{i}^{2}$ by $1$ as long as possible. Denote the resulting polynomial by $F_{\overrightarrow {a}}(x_{2}, \ldots , x_{n})$.

Prove that if no two vectors among $\overrightarrow {a}_{1}, \ldots , \overrightarrow {a}_{s} \in M$ have zero scalar product, then the polynomials $F_{\overrightarrow {a}_{1}}, \ldots , F_{\overrightarrow {a}_{s}}$ are linearly independent over $\mathbb {Q}$.

There exist $n$ and a bounded subset of $\mathbb {R}^{n}$ that cannot be partitioned into $n+1$ nonempty parts of smaller diameter.

If $t$ is prime, then among any $1 + \sum \limits_{j=0}^{t-1} \dbinom {n}{j}$ distinct subsets of an $n$-element set, each of which has $k$ elements, there are two subsets whose intersection has a number of elements divisible by $t$.

The same, except that an integer $q$ is given and is divisible by $t$'' is replaced by is congruent to $q$ modulo $t$''.

If the edge set of the graph $K_{n}$ is the union of the edge sets of $s$ edge-disjoint complete bipartite graphs, then $s \geq n-1$.

Construct a family of bipartite graphs for which this bound is attained.

$n = 2$;

$n = 4$;

$n = 8$;

$n = 16$;

$n = 12$.

Any two columns of a Hadamard matrix are orthogonal.

A matrix is a Hadamard matrix if and only if its entries are equal to $\pm 1$ and any two of its rows are orthogonal.

Hadamard matrices attain the upper bound in Hadamard's theorem 7.2.1. (Hadamard matrices got their name owing to this result.)

If there exists an $n \times n$ Hadamard matrix and $n > 2$, then $n$ is divisible by $4$.

Prove that $S_{d}$ does not depend on $d \neq 0$.

Find $S_{d}$ for each $d \in \mathbb {Z}_{p}$.

$n = 2a$ if there exists an $a \times a$ Hadamard matrix;

$n = ab$ if there exist $a \times a$ and $b \times b$ Hadamard matrices;

$n = p+1$, where $p$ is a prime number of the form $4k-1$;

$n = 2p+2$, where $p$ is a prime number of the form $4k+1$.

multiplying a row or a column by $-1$;

interchanging rows or columns.

Which of the matrices from Problem 7.2.6 are equivalent?

Every Hadamard matrix is equivalent to some normalized one.