Abstract: One of the most challenging open problems in extremal combinatorics,
proposed by Erd\H{o}s and S\'{o}s in 1971, asks to determine for every
triple of positive integers $l\leq k\leq n$ the largest subfamily
$\mathcal{A}$ of the $k$-th layer of the cube, whose interesections forbid
$l$, that is, it satisfies $|A\cap B|\neq l$ for every
$A,B\in\mathcal{A}$.

Significant progress on the Erd\H{o}s--S\'{o}s problem was made by several
authors including Frankl/Wilson (1981), Frankl/Furedi (1985) and
Frankl/R\"{o}dl (1987), and much more recently, by Ellis/Keller/Lifshitz
(2016), Keller/Lifshitz (2021) and Kupavskii/Zaharov (2022).

We shall discuss these results, and we shall outline the proof of the
following new estimate.

Let $n$ be a positive integer, let $0<p\leq p'\leq 1/2$, and let $l\leq
pn$ be a nonnegative integer. Also let $\mathcal{F},\mathcal{G}\subseteq
\{0,1\}^n$ be two families whose cross intersections forbid $l$, that is,
$|A\cap B|\neq l$ for every $A\in\mathcal{F}$ and $B\in\mathcal{G}$. Then,
setting $t:=\min\{l,pn-l\}$, we have the subgaussian bound
\[ \mu_p(\mathcal{F}) \mu_{p'}(\mathcal{G}) \leq
2\exp\Big(-\frac{t^2}{58^2 pn}\Big), \]
where $\mu_p$ and $\mu_q$ denote the $p$-biased and $q$-biased measures on
$\{0,1\}^n$, respectively.

This estimate extends the celebrated Frankl--R\"{o}dl theorem to the
sparse setting, and it is actually optimal (modulo universal constants)
for various choices of $p,q,l$.

This is joint work with Miltiadis Karamanlis.

Abstract: One of the most challenging open problems in extremal combinatorics,
proposed by Erd\H{o}s and S\'{o}s in 1971, asks to determine for every
triple of positive integers $l\leq k\leq n$ the largest subfamily
$\mathcal{A}$ of the $k$-th layer of the cube, whose interesections forbid
$l$, that is, it satisfies $|A\cap B|\neq l$ for every
$A,B\in\mathcal{A}$.

Significant progress on the Erd\H{o}s--S\'{o}s problem was made by several
authors including Frankl/Wilson (1981), Frankl/Furedi (1985) and
Frankl/R\"{o}dl (1987), and much more recently, by Ellis/Keller/Lifshitz
(2016), Keller/Lifshitz (2021) and Kupavskii/Zaharov (2022).

We shall discuss these results, and we shall outline the proof of the
following new estimate.

Let $n$ be a positive integer, let $0<p\leq p'\leq 1/2$, and let $l\leq
pn$ be a nonnegative integer. Also let $\mathcal{F},\mathcal{G}\subseteq
\{0,1\}^n$ be two families whose cross intersections forbid $l$, that is,
$|A\cap B|\neq l$ for every $A\in\mathcal{F}$ and $B\in\mathcal{G}$. Then,
setting $t:=\min\{l,pn-l\}$, we have the subgaussian bound
\[ \mu_p(\mathcal{F}) \mu_{p'}(\mathcal{G}) \leq
2\exp\Big(-\frac{t^2}{58^2 pn}\Big), \]
where $\mu_p$ and $\mu_q$ denote the $p$-biased and $q$-biased measures on
$\{0,1\}^n$, respectively.

This estimate extends the celebrated Frankl--R\"{o}dl theorem to the
sparse setting, and it is actually optimal (modulo universal constants)
for various choices of $p,q,l$.

This is joint work with Miltiadis Karamanlis.

Abstract: One of the most challenging open problems in extremal combinatorics,
proposed by Erd\H{o}s and S\'{o}s in 1971, asks to determine for every
triple of positive integers $l\leq k\leq n$ the largest subfamily
$\mathcal{A}$ of the $k$-th layer of the cube, whose interesections forbid
$l$, that is, it satisfies $|A\cap B|\neq l$ for every
$A,B\in\mathcal{A}$.

Significant progress on the Erd\H{o}s--S\'{o}s problem was made by several
authors including Frankl/Wilson (1981), Frankl/Furedi (1985) and
Frankl/R\"{o}dl (1987), and much more recently, by Ellis/Keller/Lifshitz
(2016), Keller/Lifshitz (2021) and Kupavskii/Zaharov (2022).

We shall discuss these results, and we shall outline the proof of the
following new estimate.

Let $n$ be a positive integer, let $0<p\leq p'\leq 1/2$, and let $l\leq
pn$ be a nonnegative integer. Also let $\mathcal{F},\mathcal{G}\subseteq
\{0,1\}^n$ be two families whose cross intersections forbid $l$, that is,
$|A\cap B|\neq l$ for every $A\in\mathcal{F}$ and $B\in\mathcal{G}$. Then,
setting $t:=\min\{l,pn-l\}$, we have the subgaussian bound
\[ \mu_p(\mathcal{F}) \mu_{p'}(\mathcal{G}) \leq
2\exp\Big(-\frac{t^2}{58^2 pn}\Big), \]
where $\mu_p$ and $\mu_q$ denote the $p$-biased and $q$-biased measures on
$\{0,1\}^n$, respectively.

This estimate extends the celebrated Frankl--R\"{o}dl theorem to the
sparse setting, and it is actually optimal (modulo universal constants)
for various choices of $p,q,l$.

This is joint work with Miltiadis Karamanlis.