arXiv subjects

Peter Frankl

Publications and source records attributed to Peter Frankl.

At least 19 recordsLinked to original sources

arXiv2026

Stability for Helly-type and triangle-free families

We consider $k$-graphs, $\mathcal{F}\subset \binom{[n]}{k}$, $k\geq 3$. A $k$-graph is called intersecting if any two of its edges have non-empty intersection. It is called a star if all its edges share a common vertex. The $k$-graph $\mathcal{F}$ is called Helly if all its intersecting subfamilies are stars. If the same is required only for subfamilies consisting of three edges, it is called triangle-free. It is well known that for $n\geq 3k/2$, the full star is the unique largest triangle-free family whence the largest Helly family as well. In 1984 Tuza proved the best possible bound $|\mathcal{F}|\leq \binom{n-k-1}{k-1}+\binom{n-2}{k-2}+1$ for Helly families that are not stars, albeit only for some unspecified $n>n_0(k)$. The aim of this paper is twofold. First we establish the same bound for $n>2k$. Second we show that for $n>12k^2$ the same upper bound holds for triangle-free families. It is shown as well that it is not true for $2k<n\leq 3k-4$.

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arXiv2026

Matching and intersection problems for non-trivial $r$-partite $r$-uniform hypergraphs

A central theme in extremal combinatorics is the study of the maximum number of edges in an $r$-uniform hypergraph ($r$-graph) with matching number at most $s$ (the Erdős Matching Conjecture) or with pairwise intersection at least $t$ (the $t$-intersection problem). The maximum sizes for these problems are typically achieved by trivial constructions: for the matching problem, the extremal construction consists of all edges intersecting a fixed set of $s$ vertices, while for the intersection problem, it consists of all edges containing a fixed set of $t$ vertices. In this paper, we investigate the \emph{non-trivial} $r$-partite $r$-graphs where each part is of size $n$. We determine the exact bounds for both the matching problem and the intersection problem when $n$ is sufficiently large. Furthermore, for the intersection problem, we resolve the cases $t=1$ and $t=r-2$ for all $n \ge 2$. Our results partially confirm a conjecture of Lu and Ma (``Matching Stability for 3-Partite 3-Uniform Hypergraphs.'' Journal of Graph Theory (2026)).

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arXiv2026

Towards the Erdős matching conjecture for 4-uniform hypergraphs: stability and applications

A famous conjecture of Erdős asserts that for $k\ge 3$, the maximum number of edges in an $n$-vertex $k$-uniform hypergraph without $s+1$ pairwise disjoint edges is $\max\{\binom{n}{k}-\binom{n-s}{k},\binom{sk+k-1}{k}\}$. This problem has been central in extremal combinatorics, with substantial progress in the literature, including a complete solution for $k=3$ due to the first author. In this paper, we make progress towards the $4$-uniform case, proving the conjecture for $n\ge 5s$ and sufficiently large $n$, thereby taking a first step analogous to the $3$-uniform case. The main technical contribution is a stability result of independent interest. We further apply this stability to resolve two new instances of conjectures on the minimum $d$-degree threshold for matchings in $5$- and $6$-uniform hypergraphs, in a strengthened form.

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arXiv2026

On the product of cross-intersecting families with small covering number

A central problem in extremal set theory is to determine or estimate $m(n,k,t),n>2k\geq 2t$, the maximum size of an intersecting $k$-graph and covering number at least $t$(see the paper for the definitions). For $t=1$ and $2$ the classical Erdős-Ko-Rado Theorem and the Hilton-Milner Theorem provide the answer.The complete solution for $t=3$ was only achieved recently . There are some partial results for $t=4,5$ but for the general case even to determine the asymptotic appears to be hopelessly difficult . Denoting by $\widetilde{m}(n,k,t)$ the maximum of $|\mathcal{F}||\mathcal{G}|$ for a pair of cross-intersecting $k$-graphs with covering number at least $t$, $\widetilde{m}(n,k,t)\geq {m}(n,k,t)^2$ is obvious. Pyber showed that equality holds for $t=1$. The same was shown for $t=2$ in a wide range(cf.[7]). Quite surprisingly our results show that the inequality is strict for $t\geq 3$ and for $n>n_0(k,t)$, Theorem 1.7 determines the exact value of $\widetilde{m}(n,k,t)$ for $k>2t$ and $n$ sufficiently large.

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arXiv2026

On the product of cross-intersecting families with maximal covering number

For integers $k ,\ell \geq 2$ let $m(k,\ell)$ denote the maximum of $|\mathcal{F}| |\mathcal{G}|$ where the maximum is taken over all pairs of cross-intersecting families, $\mathcal{F}$ being a $k$-graph with covering number $\ell$ and $\mathcal{G}$ a $\ell$-graph with covering number $k$ (see the paper for the definitions). Erdos and Lovasz initiated the study of the one family version. That is, they provided lower and upper bounds on the maximal size $m(k)=|\mathcal{F}|$ where $\mathcal{F}$ is an intersecting k-graph with covering number $k$. In many similar situations $m(k,k)=m(k)^2$ holds. However, as our results show $m(k,k)/m(k)^2$ is tending to infinity as $k$ grows(Th.1.5) . For $k>k_0$ we establish the exact value $m(k,k)=(k^{k-1}+k-1)^2$(Th.1.6). As to smaller values we prove $m(3,3)=121$ (Th.1.7) and determine $m(2,k) $ for all $k\geq 2$ (Th.1.8).

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arXiv2026

A product version of the Hilton-Milner Theorem II

Two families $\mathcal{F},\mathcal{G}$ of $k$-subsets of $\{1,2,\ldots,n\}$ are called {\it non-trivial cross-intersecting} if $F\cap G\neq \emptyset$ for all $F\in \mathcal{F}, G\in \mathcal{G}$ and $\cap \{F\colon F\in \mathcal{F}\}=\emptyset=\cap \{G\colon G\in\mathcal{G}\}$. In this note, we establish the product version of the Hilton-Milner Theorem for $k\geq 8$ in the full range. That is, if $\mathcal{F},\mathcal{G}\subset \binom{[n]}{k}$ are non-trivial cross-intersecting, $n\geq 2k+1$ and $k\geq 8$, then \[ |\mathcal{F}||\mathcal{G}|\leq \left(\binom{n-1}{k-1}- \binom{n-k-1}{k-1} +1\right)^2. \]

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arXiv2026

Intersecting families with covering number three II

A family $\mathcal{F}\subset \binom{[n]}{k}$ is called intersecting if $F\cap F'\neq \emptyset$ for all $F,F'\in \mathcal{F}$. The covering number of a family $\mathcal{F}$ is defined as the minimum size of $T\subset [n]$ such that $T\cap F\neq \emptyset$ for all $F\in \mathcal{F}$. In 1980, the first author proved that for sufficiently large $n$, any intersecting $k$-graph $\mathcal{F}$ with covering number at least three, satisfies $|\mathcal{F}|\leq \binom{n-1}{k-1}-\binom{n-k}{k-1}-\binom{n-k-1}{k-1}+\binom{n-2k}{k-1}+\binom{n-k-2}{k-3}+3$. There was very little progress during more than forty years but recently (cf. \cite{FW25}) with a completely different approach we proved the same result for the full range $n\geq 2k$ and $k\geq 7$. In this short paper we prove the same inequality for all the remaining cases.

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arXiv2026

The Exact Erdős-Ko-Rado Theorem for 3-wise $t$-intersecting uniform families

Let $\mathcal{F}$ be a family of $k$-element subsets of $\{1,2,\ldots,n\}$. For $t\geq 1$, we say that $\mathcal{F}$ is {\it 3-wise $t$-intersecting} if $|F_1\cap F_2\cap F_3|\geq t$ for all $F_1,F_2,F_3\in \mathcal{F}$. In the present paper, we prove that if $\mathcal{F}$ is 3-wise $t$-intersecting and $n\geq \frac{\sqrt{4t+9}-1}{2}k$, $k>t\geq 46$, then $|\mathcal{F}|\leq \binom{n-t}{k-t}$. The restriction on $n$ is asymptotically best possible. The corresponding result for non-trivial 3-wise $t$-intersecting families is obtained as well for $n\geq \frac{\sqrt{4t+9}-1}{2}k$ and $k>t\geq 55$.

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arXiv2025

On the largest degrees in intersecting hypergraphs

Let $\binom{[n]}{k}$ denote the collection of all $k$-subsets of the standard $n$-set $[n]=\{1,2,\ldots,n\}$. Let $n>2k$ and let $\mathcal{F}\subset \binom{[n]}{k}$ be an {\it intersecting} $k$-graph, i.e., $F\cap F'\neq \emptyset$ for all $F,F'\in \mathcal{F}$. The number of edges $F\in \mathcal{F}$ containing $x\in [n]$ is called the {\it degree} of $x$. Assume that $d_1\geq d_2\geq \ldots\geq d_n$ are the degrees of $\mathcal{F}$ in decreasing order. An important result of Huang and Zhao states that for $n>2k$ the minimum degree $d_n$ is at most $\binom{n-2}{k-2}$. For $n\geq 6k-9$ we strengthen this result by showing $d_{2k+1}\leq \binom{n-2}{k-2}$. As to the second and third largest degrees we prove the best possible bound $d_3\leq d_2\leq \binom{n-2}{k-2}+\binom{n-3}{k-2}$ for $n>2k$. Several more best possible results of a similar nature are established.

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arXiv2025

On the Matching Problem in Random Hypergraphs

We study a variant of the Erdős Matching Problem in random hypergraphs. Let $\mathcal{K}_p(n,k)$ denote the Erdős-Rényi random $k$-uniform hypergraph on $n$ vertices where each possible edge is included with probability $p$. We show that when $n\gg k^{2}s$ and $p$ is not too small, with high probability, the maximum number of edges in a sub-hypergraph of $\mathcal{K}_p(n,k)$ with matching number $s$ is obtained by the trivial sub-hypergraphs, i.e. the sub-hypergraph consisting of all edges containing at least one vertex in a fixed set of $s$ vertices.

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arXiv2025

Intersecting Families of Spanning Trees

A family $\mathcal{F}$ of spanning trees of the complete graph on $n$ vertices $K_n$ is \emph{$t$-intersecting} if any two members have a forest on $t$ edges in common. We prove an Erdős--Ko--Rado result for $t$-intersecting families of spanning trees of $K_n$. In particular, we show there exists a constant $C > 0$ such that for all $n \geq C (\log n) t$ the largest $t$-intersecting families are the families consisting of all trees that contain a fixed set of $t$ disjoint edges (as well as the stars on $n$ vertices for $t = 1$). The proof uses the spread approximation technique in conjunction with the Lopsided Lovász Local Lemma.

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arXiv2025

The overflow in the Katona Theorem

Let $n>2r>0$ be integers. We consider families $\mathcal{F}$ of subsets of an $n$-element set, in which the union of any two members has size at most $2r$. One of our results states that for $n\geq 6r$ the number of members of size exceeding $r$ in $\mathcal{F}$ is at most $\binom{n-2}{r-1}$. Another result shows that for $n>3.5r$ the number of sets of size at least $r$ is at most $\binom{n}{r}$. Both bounds are best possible and the latter sharpens the classical Katona Theorem. Similar results are proved for the odd case of the Katona Theorem as well.

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arXiv2025

On resilient hypergraphs

The matching number of a $k$-graph is the maximum number of pairwise disjoint edges in it. The $k$-graph is called $t$-resilient if omitting $t$ vertices never decreases its matching number. The complete $k$-graph on $sk+k-1$ vertices has matching number $s$ and it is easily seen to be $(k-1)$-resilient. We conjecture that this is maximal for $k=3$ and $s$ arbitrary. The main result verifies this conjecture for $s=2$. Then Theorem 1.9 provides a considerable improvement on the known upper bounds for $s\geq 3$.

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arXiv2025

The Hajnal--Rothschild problem

For a family $\mathcal F$ define $ν(\mathcal F,t)$ as the largest $s$ for which there exist $A_1,\ldots, A_{s}\in \mathcal F$ such that for $i\ne j$ we have $|A_i\cap A_j|< t$. What is the largest family $\mathcal F\subset{[n]\choose k}$ with $ν(\mathcal F,t)\le s$? This question goes back to a paper Hajnal and Rothschild from 1973. We show that, for some absolute $C$ and $n>2k+Ct^{4/5}s^{1/5}(k-t)\log_2^4n$, $n>2k+Cs(k-t)\log_2^4 n$ the largest family with $ν(\mathcal F,t)\le s$ has the following structure: there are sets $X_1,\ldots, X_s$ of sizes $t+2x_1,\ldots, t+2x_s$, such that for any $A\in \mathcal F$ there is $i\in [s]$ such that $|A\cap X_i|\ge t+x_i$. That is, the extremal constructions are unions of the extremal constructions in the Complete $t$-Intersection Theorem. For the proof, we enhance the spread approximation technique of Zakharov and the second author. In particular, we introduce the idea of iterative spread approximation.

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arXiv2024

Intersecting families with covering number three

We consider $k$-graphs on $n$ vertices, that is, $\mathcal{F}\subset \binom{[n]}{k}$. A $k$-graph $\mathcal{F}$ is called intersecting if $F\cap F'\neq \emptyset$ for all $F,F'\in \mathcal{F}$. In the present paper we prove that for $k\geq 7$, $n\geq 2k$, any intersecting $k$-graph $\mathcal{F}$ with covering number at least three, satisfies $|\mathcal{F}|\leq \binom{n-1}{k-1}-\binom{n-k}{k-1}-\binom{n-k-1}{k-1}+\binom{n-2k}{k-1}+\binom{n-k-2}{k-3}+3$, the best possible upper bound which was proved in \cite{F80} subject to exponential constraints $n>n_0(k)$.

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arXiv2024

On the $C$-diversity of intersecting hypergraphs

Let $\mathcal{F}\subset \binom{X}{k}$ be a family consisting of $k$-subsets of the $n$-set $X$. Suppose that $\mathcal{F}$ is intersecting, i.e., $F\cap F'\neq \emptyset$ for all $F,F'\in \mathcal{F}$. Let $Δ(\mathcal{F})$ be the maximum degree of $\mathcal{F}$. For a constant $C\geq 1$ the $C$-diversity, $γ_C(\mathcal{F})$ is defined as $|\mathcal{F}|-CΔ(\mathcal{F})$. Define $\mathcal{F}_{123} =\left\{F\in \binom{X}{k}\colon |F\cap \{1,2,3\}|=2\right\}$. It has $C$-diversity $(3-2C)\binom{n-3}{k-2}$. The main result shows that for $1< C<\frac{3}{2}$ and $n\geq \frac{42}{3-2C}k$, $γ_C(\mathcal{F})\leq γ_C(\mathcal{F}_{123})$ with equality if and only if $\mathcal{F}$ is isomorphic to $\mathcal{F}_{123}$. For the case of ordinary diversity $(C=1)$ a strong stability is proven.

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arXiv2024

The maximum sturdiness of intersecting families

Given a family $\mathcal{F}\subset 2^{[n]}$ and $1\leq i\neq j\leq n$, we use $\mathcal{F}(\bar{i},j)$ to denote the family $\{F\setminus \{j\}\colon F\in \mathcal{F},\ F\cap \{i,j\}=\{j\}\}$. The sturdiness of $\mathcal{F}$ is defined as the minimum $|\mathcal{F}(\bar{i},j)|$ over all $i,j\in [n]$ with $i\neq j$. It has a very natural algebraic definition as well. In the present paper, we consider the maximum sturdiness of $k$-uniform intersecting families, $k$-uniform $t$-intersecting families and non-uniform $t$-intersecting families. One of the main results shows that for $n\geq 36(k+6)$, an intersecting family $\mathcal{F}\subset \binom{[n]}{k}$ has sturdiness at most $\binom{n-4}{k-3}$, which is best possible.

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arXiv2024

Non-trivial $r$-wise agreeing families

A family of subsets of $[n]$ is $r$-wise agreeing if for any $r$ sets from the family there is an element $x$ that is either contained in all or contained in none of the $r$ sets. The study of such families is motivated by questions in discrete optimization. In this paper, we determine the size of the largest non-trivial $r$-wise agreeing family. This can be seen as a generalization of the classical Brace-Daykin theorem.

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