We consider the length of ordered loose paths in the random r-uniform hypergraph H = H-(r)(n, p). An ordered loose path is a sequence of edges E-1, E-2, ... , E-& ell; where max{j is an element of E-i} = min{j is an element of Ei+1} for 1 <= i < & ell;. We establish fairly tight bounds on the length of the longest ordered loose path in H that hold with high probability. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

A -coloring of is a coloring of the edges of such that every -clique has at least distinct colors among its edges. The generalized Ramsey number is the minimum number of colors such that has a -coloring. Gomez-Leos, Heath, Parker, Schweider and Zerbib recently proved . Here we prove an asymptotically matching upper bound.

An ordered r-uniform matching of size n is a collection of n pairwise disjoint r-subsets of a linearly ordered set of rn vertices. For n=2, such a matching is called an r-pattern, as it represents one of 122rr ways two disjoint edges may intertwine. Given a set 𝒫 of r-patterns, a 𝒫-clique is a matching with all pairs of edges belonging to 𝒫. In this paper we determine the order of magnitude of the size of a largest 𝒫-clique in a random ordered r-uniform matching for several sets 𝒫, including all sets of size |𝒫|≤2 and the set ℛ^(r) of all 2^r-1 r-partite r-patterns.

ABSTRACT A ‐ coloring of is a coloring of the edges of such that every ‐clique has at least distinct colors among its edges. The generalized Ramsey number is the minimum number of colors such that has a ‐coloring. Gomez‐Leos, Heath, Parker, Schweider and Zerbib recently proved . Here we prove an asymptotically matching upper bound.

The triangle packing number ν(G) of a graph G is the maximum size of a set of edge-disjoint triangles in G. Tuza conjectured that in any graph G there exists a set of at most 2ν(G) edges intersecting every triangle in G. We show that Tuza's conjecture holds in the random geometric graph for a large range of densities. We also study the problem of covering almost all edges of the random geometric graph with edge-disjoint copies of some fixed graph F. In particular, we show the existence of almost-perfect packings for an infinite family of F, and state some negative results as well.

We study the question of the least number of random edges that need to be added to a Pósa-Seymour graph, that is, a graph with minimum degree exceeding k/k+1n, to secure the existence of the m-th power of a Hamiltonian cycle, m>k. It turns out that, depending on k and m, this quantity may be captured by two types of thresholds, with one of them, called over-threshold, becoming dominant for large m. Indeed, for each k≥2 and m>m_0(k), we establish asymptotically tight lower and upper bounds on the over-thresholds (provided they exist) and show that for infinitely many instances of m the two bounds coincide. In addition, we also determine the thresholds for some small values of k and m.

The Erdős-Gyárfás number f(n, p, q) is the smallest number of colors needed to color the edges of the complete graph K_n so that all of its p-clique spans at least q colors. In this paper we improve the best known upper bound on f(n, p, q) for many fixed values of p, q and large n. Our proof uses a randomized coloring process, which we analyze using the so-called differential equation method to establish dynamic concentration.

An ordered r-uniform matching of size n is a collection of n pairwise disjoint r-subsets of a linearly ordered set of rn vertices. For n=2, such a matching is called an r-pattern, as it represents one of 122rr ways two disjoint edges may intertwine. Given a set 𝒫 of r-patterns, a 𝒫-clique is a matching with all pairs of edges order-isomorphic to a member of 𝒫. In this paper we are interested in the size of a largest 𝒫-clique in a random ordered r-uniform matching selected uniformly from all such matchings on a fixed vertex set [rn]. We determine this size (up to multiplicative constants) for several sets 𝒫, including all sets of size |𝒫|≤2, the set ℛ^(r) of all r-partite patterns, as well as sets 𝒫 enjoying a Boolean-like, symmetric structure.

For r, n 2, an ordered r-uniform matching of size n is an r-uniform hypergraph on a linearly ordered vertex set V, with |V | = rn, consisting of n pairwise disjoint edges. There are 1 ( 2r ) different ways two edges may intertwine, called here 2 r patterns. Among them we identify 3 r - 1 collectable patterns P, which have the potential of appearing in arbitrarily large quantities called P-cliques. We prove an Erd & odblac;s-Szekeres type result guaranteeing in every ordered r-uniform matching the presence of a P-clique of a prescribed size, for some collectable pattern P. In particular, in the diagonal case, one of the P-cliques must be of size (n31-r// n 3 1 - r // Omega . In addition, for each collectable pattern P we show that the largest size of a P-clique in a random ordered runiform matching of size n is, with high probability, Theta ( n 1/r ) . (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

The generalized Ramsey number $f(n, p, q)$ is the smallest number of colors needed to color the edges of the complete graph $K_n$ so that every $p$-clique spans at least $q$ colors. Erdős and Gyárfás showed that $f(n, p, q)$ grows linearly in $n$ when $p$ is fixed and $q=q_{\text{lin}}(p):=\binom p2-p+3$. Similarly they showed that $f(n, p, q)$ is quadratic in $n$ when $p$ is fixed and $q=q_{\text{quad}}(p):=\binom p2-\frac p2+2$. In this note we improve on the known estimates for $f(n, p, q_{\text{lin}})$ and $f(n, p, q_{\text{quad}})$. Our proofs involve establishing a significant strengthening of a previously known connection between $f(n, p, q)$ and another extremal problem first studied by Brown, Erdős and Sós, as well as building on some recent progress on this extremal problem by Delcourt and Postle and by Shangguan. Also, our upper bound on $f(n, p, q_{\text{lin}})$ follows from an application of the recent forbidden submatchings method of Delcourt and Postle.

An ordered matching of size $n$ is a graph on a linearly ordered vertex set $V$, $|V|=2n$, consisting of $n$ pairwise disjoint edges. There are three different ordered matchings of size two on $V=\{1,2,3,4\}$: an alignment $\{1,2\},\{3,4\}$, a nesting $\{1,4\},\{2,3\}$, and a crossing $\{1,3\},\{2,4\}$. Accordingly, there are three basic homogeneous types of ordered matchings (with all pairs of edges arranged in the same way) which we call, respectively, lines, stacks, and waves. We prove an Erdős-Szekeres type result guaranteeing in every ordered matching of size $n$ the presence of one of the three basic sub-structures of a given size. In particular, one of them must be of size at least $n^{1/3}$. We also investigate the size of each of the three sub-structures in a random ordered matching. Additionally, the former result is generalized to $3$-uniform ordered matchings. Another type of unavoidable patterns we study are twins, that is, pairs of order-isomorphic, disjoint sub-matchings. By relating to a similar problem for permutations, we prove that the maximum size of twins that occur in every ordered matching of size $n$ is $O\left(n^{2/3}\right)$ and $\Omega\left(n^{3/5}\right)$. We conjecture that the upper bound is the correct order of magnitude and confirm it for almost all matchings. In fact, our results for twins are proved more generally for $r$-multiple twins, $r\ge2$.

An ordered $r$-matching of size $n$ is an $r$-uniform hypergraph on a linearly ordered set of vertices, consisting of $n$ pairwise disjoint edges. Two ordered $r$-matchings are isomorphic if there is an order-preserving isomorphism between them. A pair of twins in an ordered $r$-matching is formed by two vertex disjoint isomorphic sub-matchings. Let $t^{(r)}(n)$ denote the maximum size of twins one may find in every ordered $r$-matching of size $n$. By relating the problem to that of largest twins in permutations and applying some recent Erd\H{o}s-Szekeres-type results for ordered matchings, we show that $t^{(r)}(n)=\Omega\left(n^{\frac{3}{5\cdot(2^{r-1}-1)}}\right)$ for every fixed $r\geqslant 2$. On the other hand, $t^{(r)}(n)=O\left(n^{\frac{2}{r+1}}\right)$, by a simple probabilistic argument. As our main result, we prove that, for almost all ordered $r$-matchings of size $n$, the size of the largest twins achieves this bound.

Let M be an ordered matching of size n, that is, a partition of the set [2n] into 2-element subsets. The sock number of M is the maximum size of a sub-matching of M in which all left-ends of the edges precede all the right-ends (such matchings are also called bipartite). The name of this parameter comes from an amusing "real-life" problem posed by Bosek, concerning an on-line pairing of randomly picked socks from a drying machine. Answering one of Bosek's questions we prove that the sock number of a random matching of size n is asymptotically equal to n/2. Moreover, we prove that the expected average number of socks waiting for their match during the whole process is equal to 2n+1/6. Analogous results are obtained if socks come not in pairs, but in sets of size r≥ 2, which corresponds to a similar problem for random ordered r-matchings. We also attempt to enumerate matchings with a given sock number.

For a graph G, let τ (G) be the maximum number of colors such that there exists an edge-coloring of G with no two color classes being isomorphic. We investigate the behavior of τ (G) when G=G(n, p) is the classical Erdős-Rényi random graph.

A (4, , 5)-coloring of K n is an edge-coloring of K n where every 4-clique spans at least five colors. We show that there exist (4, , 5)-colorings of K n using 65 n + o ( n ) colors. This settles a disagreement between Erd & odblac;s and Gy & aacute;rf & aacute;s reported in their 1997 paper. Our construction uses a randomized process which we analyze using the so-called differential equation method to establish dynamic concentration. In particular, our coloring process uses random triangle removal, a process first introduced by Bollob & aacute;s and Erd & odblac;s, and analyzed by Bohman, Frieze and Lubetzky. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

A (p, q)-coloring of K_n is a coloring of the edges of K_n such that every p-clique has at least q distinct colors among its edges. The generalized Ramsey number f(n, p, q) is the minimum number of colors such that K_n has a (p, q)-coloring. Gomez-Leos, Heath, Parker, Schweider and Zerbib recently proved f(n, 5, 8) ≥6/7 (n-1). Here we prove an asymptotically matching upper bound.

We study the existence of powers of Hamiltonian cycles in graphs with large minimum degree to which some additional edges have been added in a random manner. It follows from the theorems of Dirac and of Komlós, Sarközy, and Szemerédi that for every k ≥ 1 and sufficiently large n already the minimum degree δ(G)≥kk+1n for an n‐vertex graph G alone suffices to ensure the existence of a kth power of a Hamiltonian cycle. Here we show that under essentially the same degree assumption the addition of just O(n) random edges ensures the presence of the (k + 1)st power of a Hamiltonian cycle with probability close to one.

By $r$-twins in a permutation we mean a collection of $r$ pairwise disjoint subsequences with the same relative order. The length of $r$-twins is defined as the length of any single subsequence in the collection. Let $t^{(r)}(n)$ denote the largest $k$ such that every permutation of length $n$ contains $r$-twins of length $k$. We prove that $t^{(r)}(n)=O\left(n^{\frac r{2r-1}}\right)$ and $t^{(r)}(n)=\Omega\left( n^{\frac{R}{2R-1}} \right)$, where $R=\binom{2r-1}r$. We conjecture that the upper bound brings the correct order of magnitude of $t^{(r)}(n)$ and support this conjecture by proving that it holds for almost all permutations. Our work generalizes previous studies of the case $r=2$.

Twins in a finite word are formed by a pair of identical subwords placed at disjoint sets of positions. We investigate the maximum length of twins in a random word over a k-letter alphabet. The obtained lower bounds for small values of k significantly improve the best estimates known in the deterministic case. Bukh and Zhou in 2016 showed that every ternary word of length n contains twins of length at least 0.34n. Our main result states that in a random ternary word of length n, with high probability, one can find twins of length at least 0.41n. In the general case of alphabets of size k⩾ 3 we obtain analogous lower bounds of the form 1.64/k+1n which are better than the known deterministic bounds for k⩽ 354 . In addition, we present similar results for multiple twins in random words.

A (4, 5)-coloring of K_n is an edge-coloring of K_n where every 4-clique spans at least five colors. We show that there exist (4, 5)-colorings of K_n using 5/6 n + o(n) colors. This settles a disagreement between Erdős and Gyárfás reported in their 1997 paper. Our construction uses a randomized process which we analyze using the so-called differential equation method to establish dynamic concentration. In particular, our coloring process uses random triangle removal, a process first introduced by Bollobás and Erdős, and analyzed by Bohman, Frieze and Lubetzky.