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Peter Frankl

Publications and source records attributed to Peter Frankl.

At least 55 records · Page 3Linked to original sources
arXiv2022

Exchange properties of finite set-systems

In a recent breakthrough, Adiprasito, Avvakumov, and Karasev constructed a triangulation of the $n$-dimensional real projective space with a subexponential number of vertices. They reduced the problem to finding a small downward closed set-system $\cal F$ covering an $n$-element ground set which satisfies the following condition: For any two disjoint members $A, B\in\cal F$, there exist $a\in A$ and $b\in B$ such that either $B\cup\{a\}\in\cal F$ and $A\cup\{b\}\setminus\{a\}\in\cal F$, or $A\cup\{b\}\in\cal F$ and $B\cup\{a\}\setminus\{b\}\in\cal F$. Denoting by $f(n)$ the smallest cardinality of such a family $\cal F$, they proved that $f(n)<2^{O(\sqrt{n}\log n)}$, and they asked for a nontrivial lower bound. It turns out that the construction of Adiprasito et al. is not far from optimal; we show that $2^{(1.42+o(1))\sqrt{n}}\le f(n)\le 2^{(1+o(1))\sqrt{2n\log n}}$. We also study a variant of the above problem, where the condition is strengthened by also requiring that for any two disjoint members $A, B\in\cal F$ with $|A|>|B|$, there exists $a\in A$ such that $B\cup\{a\}\in\cal F$. In this case, we prove that the size of the smallest $\cal F$ satisfying this stronger condition lies between $2^{Ω(\sqrt{n}\log n)}$ and $2^{O(n\log\log n/\log n)}$.

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arXiv2022

Best possible bounds on the number of distinct differences in intersecting families

For a family $\mathcal F$, let $\mathcal D(\mathcal F)$ stand for the family of all sets that can be expressed as $F\setminus G$, where $F,G\in \mathcal F$. A family $\mathcal F$ is intersecting if any two sets from the family have non-empty intersection. In this paper, we study the following question: what is the maximum of $|\mathcal D(\mathcal F)|$ for an intersecting family of $k$-element sets? Frankl conjectured that the maximum is attained when $\mathcal F$ is the family of all sets containing a fixed element. We show that this holds if $n \ge 50k\ln k$ and $k \ge 50$. At the same time, we provide a counterexample for $n< 4k$.

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arXiv2022

Intersections and Distinct Intersections in Cross-intersecting Families

Let $\mathcal{F},\mathcal{G}$ be two cross-intersecting families of $k$-subsets of $\{1,2,\ldots,n\}$. Let $\mathcal{F}\wedge \mathcal{G}$, $\mathcal{I}(\mathcal{F},\mathcal{G})$ denote the families of all intersections $F\cap G$ with $F\in \mathcal{F},G\in \mathcal{G}$, and all distinct intersections $F\cap G$ with $F\neq G, F\in \mathcal{F},G\in \mathcal{G}$, respectively. For a fixed $T\subset \{1,2,\ldots,n\}$, let $\mathcal{S}_T$ be the family of all $k$-subsets of $\{1,2,\ldots,n\}$ containing $T$. In the present paper, we show that $|\mathcal{F}\wedge \mathcal{G}|$ is maximized when $\mathcal{F}=\mathcal{G}=\mathcal{S}_{\{1\}}$ for $n\geq 2k^2+8k$, while surprisingly $|\mathcal{I}(\mathcal{F}, \mathcal{G})|$ is maximized when $\mathcal{F}=\mathcal{S}_{\{1,2\}}\cup \mathcal{S}_{\{3,4\}}\cup \mathcal{S}_{\{1,4,5\}}\cup \mathcal{S}_{\{2,3,6\}}$ and $\mathcal{G}=\mathcal{S}_{\{1,3\}}\cup \mathcal{S}_{\{2,4\}}\cup \mathcal{S}_{\{1,4,6\}}\cup \mathcal{S}_{\{2,3,5\}}$ for $n\geq 100k^2$. The maximum number of distinct intersections in a $t$-intersecting family is determined for $n\geq 3(t+2)^3k^2$ as well.

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arXiv2022

Graphs without rainbow triangles

Let F,G,H be three graphs on the same n vertices. We consider the maximum of the sum and product of the number of their edges subject to the condition in the title.

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arXiv2022

Perfect matchings in down-sets

In this paper, we show that, given two down-sets (simplicial complexes) there is a matching between them that matches disjoint sets and covers the smaller of the two down-sets. This result generalizes an unpublished result of Berge from circa 1980. The result has nice corollaries for cross-intersecting families and Chvátal's conjecture. More concretely, we show that Chvátal's conjecture is true for intersecting families with covering number $2$. A family $\mathcal F\subset 2^{[n]}$ is intersection-union (IU) if for any $A,B\in\mathcal F$ we have $1\le |A\cap B|\le n-1$. Using the aforementioned result, we derive several exact product- and sum-type results for IU-families.

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arXiv2021

On well-connected sets of strings

Given $n$ pairwise disjoint sets $X_1,\ldots, X_n$, we call the elements of $S=X_1\times\ldots\times X_n$ strings. A nonempty set of strings $W\subseteq S$ is said to be well-connected if for every $v\in W$ and for every $i\, (1\le i\le n)$, there is another element $v'\in W$ which differs from $v$ only in its $i$th coordinate. We prove a conjecture of Yaokun Wu and Yanzhen Xiong by showing that every set of more than $\prod_{i=1}^n|X_i|-\prod_{i=1}^n(|X_i|-1)$ strings has a well-connected subset. This bound is tight.

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arXiv2021

A Variant of the VC-dimension with Applications to Depth-3 Circuits

We introduce the following variant of the VC-dimension. Given $S \subseteq \{0, 1\}^n$ and a positive integer $d$, we define $\mathbb{U}_d(S)$ to be the size of the largest subset $I \subseteq [n]$ such that the projection of $S$ on every subset of $I$ of size $d$ is the $d$-dimensional cube. We show that determining the largest cardinality of a set with a given $\mathbb{U}_d$ dimension is equivalent to a Turán-type problem related to the total number of cliques in a $d$-uniform hypergraph. This allows us to beat the Sauer--Shelah lemma for this notion of dimension. We use this to obtain several results on $Σ_3^k$-circuits, i.e., depth-$3$ circuits with top gate OR and bottom fan-in at most $k$: * Tight relationship between the number of satisfying assignments of a $2$-CNF and the dimension of the largest projection accepted by it, thus improving Paturi, Saks, and Zane (Comput. Complex. '00). * Improved $Σ_3^3$-circuit lower bounds for affine dispersers for sublinear dimension. Moreover, we pose a purely hypergraph-theoretic conjecture under which we get further improvement. * We make progress towards settling the $Σ_3^2$ complexity of the inner product function and all degree-$2$ polynomials over $\mathbb{F}_2$ in general. The question of determining the $Σ_3^3$ complexity of IP was recently posed by Golovnev, Kulikov, and Williams (ITCS'21).

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arXiv2021

The Erdős Matching Conjecture and concentration inequalities

More than 50 years ago, Erd\H os asked the following question: what is the maximum size of a family $\mathcal F$ of $k$-element subsets of an $n$-element set if it has no $s+1$ pairwise disjoint sets? This question attracted a lot of attention recently, in particular, due to its connection to various combinatorial, probabilistic and theoretical computer science problems. Improving the previous best bound due to the first author, we prove that $|\mathcal F|\le {n\choose k}-{n-s\choose k}$, provided $n\ge \frac 53sk -\frac 23 s$ and $s$ is sufficiently large. We derive several corollaries concerning Dirac thresholds and deviations of sums of random variables. We also obtain several related results.

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arXiv2021

On the maximum number of distinct intersections in an intersecting family

For $n > 2k \geq 4$ we consider intersecting families $\mathcal F$ consisting of $k$-subsets of $\{1, 2, \ldots, n\}$. Let $\mathcal I(\mathcal F)$ denote the family of all distinct intersections $F \cap F'$, $F \neq F'$ and $F, F'\in \mathcal F$. Let $\mathcal A$ consist of the $k$-sets $A$ satisfying $|A \cap \{1, 2, 3\}| \geq 2$. We prove that for $n \geq 50 k^2$ $|\mathcal I(\mathcal F)|$ is maximized by $\mathcal A$.

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arXiv2021

On the sum of sizes of overlapping families

Let $\mathcal{A}_1,\ldots,\mathcal{A}_m$ be families of $k$-subsets of an $n$-set. Suppose that one cannot choose pairwise disjoint edges from $s+1$ distinct families. Subject to this condition we investigate the maximum of $|\mathcal{A}_1|+\ldots+|\mathcal{A}_m|$. Note that the subcase $m=s+1$, $\mathcal{A}_1=\ldots=\mathcal{A}_m$ is the Erdős Matching Conjecture, one of the most important open problems in extremal set theory. We provide some upper bounds, a general conjecture and its solution for the range $n\geq 4k^2s$.

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arXiv2021

Almost intersecting families

Let $n > k > 1$ be integers, $[n] = \{1, \ldots, n\}$. Let $\mathcal F$ be a family of $k$-subsets of~$[n]$. The family $\mathcal F$ is called intersecting if $F \cap F' \neq \emptyset$ for all $F, F' \in \mathcal F$. It is called almost intersecting if it is not intersecting but to every $F \in \mathcal F$ there is at most one $F'\in \mathcal F$ satisfying $F \cap F' = \emptyset$. Gerbner et al. proved that if $n \geq 2k + 2$ then $|\mathcal F| \leq {n - 1\choose k - 1}$ holds for almost intersecting families. The main result implies the considerably stronger and best possible bound $|\mathcal F| \leq {n - 1\choose k - 1} - {n - k - 1\choose k - 1} + 2$ for $n > (2 + o(1))k$.

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arXiv2021

Intersection theorems for triangles

Given a family of sets on the plane, we say that the family is intersecting if for any two sets from the family their interiors intersect. In this paper, we study intersecting families of triangles with vertices in a given set of points. In particular, we show that if a set $P$ of $n$ points is in convex position, then the largest intersecting family of triangles with vertices in $P$ contains at most $(\frac{1}{4}+o(1))\binom{n}{3}$ triangles.

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arXiv2020

On the Maximum Number of Edges in Hypergraphs with Fixed Matching and Clique Number

For a $k$-graph $\mathcal{F}\subset \binom{[n]}{k}$, the clique number of $\mathcal{F}$ is defined to be the maximum size of a subset $Q$ of $[n]$ with $\binom{Q}{k}\subset \mathcal{F}$. In the present paper, we determine the maximum number of edges in a $k$-graph on $[n]$ with matching number at most $s$ and clique number at least $q$ for $n\geq 8k^2s$ and for $q \geq (s+1)k-l$, $n\leq (s+1)k+s/(3k)-l$. Two special cases that $q=(s+1)k-2$ and $k=2$ are solved completely.

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arXiv2020

Beyond the Erdős Matching Conjecture

A family $\mathcal F\subset {[n]\choose k}$ is $U(s,q)$ of for any $F_1,\ldots, F_s\in \mathcal F$ we have $|F_1\cup\ldots\cup F_s|\le q$. This notion generalizes the property of a family to be $t$-intersecting and to have matching number smaller than $s$. In this paper, we find the maximum $|\mathcal F|$ for $\mathcal F$ that are $U(s,q)$, provided $n>C(s,q)k$ with moderate $C(s,q)$. In particular, we generalize the result of the first author on the Erdős Matching Conjecture and prove a generalization of the Erdős-Ko-Rado theorem, which states that for $n> s^2k$ the largest family $\mathcal F\subset {[n]\choose k}$ with property $U(s,s(k-1)+1)$ is the star and is in particular intersecting. (Conversely, it is easy to see that any intersecting family in ${[n]\choose k}$ is $U(s,s(k-1)+1)$.) We investigate the case $k=3$ more thoroughly, showing that, unlike in the case of the Erdős Matching Conjecture, in general there may be $3$ extremal families.

math.CO↗
arXiv2020

On non-empty cross-intersecting families

Let $2^{[n]}$ and $\binom{[n]}{i}$ be the power set and the class of all $i$-subsets of $\{1,2,\cdots,n\}$, respectively. We call two families $\mathscr{A}$ and $\mathscr{B}$ cross-intersecting if $A\cap B\neq \emptyset$ for any $A\in \mathscr{A}$ and $B\in \mathscr{B}$. In this paper we show that, for $n\geq k+l,l\geq r\geq 1,c>0$ and $\mathscr{A}\subseteq \binom{[n]}{k},\mathscr{B}\subseteq \binom{[n]}{l}$, if $\mathscr{A}$ and $\mathscr{B}$ are cross-intersecting and $\binom{n-r}{l-r}\leq|\mathscr{B}|\leq \binom{n-1}{l-1}$, then $$|\mathscr{A}|+c|\mathscr{B}|\leq \max\left\{\binom{n}{k}-\binom{n-r}{k}+c\binom{n-r}{l-r},\ \binom{n-1}{k-1}+c\binom{n-1}{l-1}\right\}$$ and the families $\mathscr{A}$ and $\mathscr{B}$ attaining the upper bound are also characterized. This generalizes the corresponding result of Hilton and Milner for $c=1$ and $r=k=l$, and implies a result of Tokushige and the second author (Theorem 1.3).

math.CO↗
arXiv2020

Simple juntas for shifted families

We say that a family $\mathcal F$ of $k$-element sets is a {\it $j$-junta} if there is a set $J$ of size $j$ such that, for any $F$, its presence in $\mathcal F$ depends on its intersection with $J$ only. Approximating arbitrary families by $j$-juntas with small $j$ is a recent powerful technique in extremal set theory. The weak point of all known junta approximation results is that they work in the range $n>Ck$, where $C$ is an extremely fast growing function of the input parameters, such as the quality of approximation or the number of families we simultaneously approximate. We say that a family $\mathcal F$ is {\it shifted} if for any $F=\{x_1,\ldots, x_k\}\in \mathcal F$ and any $G =\{y_1,\ldots, y_k\}$ such that $y_i\le x_i$, we have $G\in \mathcal F$. For many extremal set theory problems, including the Erd\H os Matching Conjecture, or the Complete $t$-Intersection Theorem, it is sufficient to deal with shifted families only. In this paper, we present very general approximation by juntas results for shifted families with explicit (and essentially linear) dependency on the input parameters. The results are best possible up to some constant factors. Moreover, they give meaningful statements for almost all range of values of $n$. The proofs are shorter than the proofs of the previous approximation by juntas results and are completely self-contained. As an application of our junta approximation, we give a nearly-linear bound for the multi-family version of the Erd\H os Matching Conjecture. More precisely, we prove the following result. Let $n\ge 12sk\log(e^2s)$ and suppose that the families $\mathcal F_1,\ldots, \mathcal F_s\subset {[n]\choose k}$ do not contain $F_1\in\mathcal F_1,\ldots, F_s\in \mathcal F_s$ such that $F_i$'s are pairwise disjoint. Then $\min_{i}|\mathcal F_i|\le {n\choose k}-{n-s+1\choose k}.$

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arXiv2020

Analogues of Katona's and Milner's Theorems for two families

Let $n>s>0$ be integers, $X$ an $n$-element set and $\mathscr{A}, \mathscr{B}\subset 2^X$ two families. If $|A\cup B|\le s$ for all $A\in\mathscr{A}, B\in \mathscr{B}$, then $\mathscr{A}$ and $\mathscr{B}$ are called cross $s$-union. Assuming that neither $\mathscr{A}$ nor $\mathscr{B}$ is empty, we prove several best possible bounds. In particular, we show that $|\mathscr{A}|+|\mathscr{B}|\le 1+\sum\limits_{0\le i\le s}{{n}\choose{i}}$. Supposing $n\ge 2s$ and $\mathscr{A},\mathscr{B}$ are antichains, we show that $|\mathscr{A}|+|\mathscr{B}|\le {{n}\choose{1}}+{{n}\choose{s-1}}$ unless $\mathscr{A}=\{\emptyset\}$ or $\mathscr{B}=\{\emptyset\}$. An analogous result for three families is established as well.

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