An edge coloring of hypergraph H is a function such that holds for any pair of intersecting edges . The minimum number of colors in edge colorings of H is called the chromatic index of H and is ...An edge coloring of hypergraph H is a function such that holds for any pair of intersecting edges . The minimum number of colors in edge colorings of H is called the chromatic index of H and is denoted by . Erdös, Faber and Lovász proposed a famous conjecture that holds for any loopless linear hypergraph H with n vertices. In this paper, we show that is true for gap-restricted hypergraphs. Our result extends a result of Alesandroni in 2021.展开更多
Erdosa and Sós conjectured in 1963 that if G is a graph o ofof ordeq >1/2p(k - 1), then G contains every tree of size k. It is shown in this paper that the conjecture is true if the complement G of G contains ...Erdosa and Sós conjectured in 1963 that if G is a graph o ofof ordeq >1/2p(k - 1), then G contains every tree of size k. It is shown in this paper that the conjecture is true if the complement G of G contains no a copy of K3 as an induced subgraph of G.展开更多
Let Xn,n ≥ 1, be a sequence of independent random variables satisfying P(Xn = 0) = 1 - P(Xn = an) = 1 - 1/Pn, where an,n ≥ 1, is a sequence of real numbers, and Pn is the nth prime,set FN(x) = P (N Xn ≤ x). The aut...Let Xn,n ≥ 1, be a sequence of independent random variables satisfying P(Xn = 0) = 1 - P(Xn = an) = 1 - 1/Pn, where an,n ≥ 1, is a sequence of real numbers, and Pn is the nth prime,set FN(x) = P (N Xn ≤ x). The authors investigate a conjecture of Erdos in probabilistic number theory and show that in order for the sequence FN to be weakly convergent, it is both sufficient and necessary that there exist three numbers X0 and X1 < X2 such that limsup(FN(X2) - FN(X1)) > 0 holds, and Lo = N→ ∞ lim FN(X0) exists. Moreover, the authors point out that they can also obtain the same result in the weakened case of lim inf P(Xn = 0) > 0.展开更多
Aharoni and Howard and,independently,Huang et al.(2012)proposed the following rainbow version of the Erd os matching conjecture:For positive integers n,k and m with n≥km,if each of the families F1,……,Fm⊆([n]k)has s...Aharoni and Howard and,independently,Huang et al.(2012)proposed the following rainbow version of the Erd os matching conjecture:For positive integers n,k and m with n≥km,if each of the families F1,……,Fm⊆([n]k)has size more than max{(n k)−(n-m+1 k);(km-1 k)},then there exist pairwise disjoint subsets e1,……,em such that ei∈Fi for all i∈[m].We prove that there exists an absolute constant n0 such that this rainbow version holds for k=3 and n≥n_(0).We convert this rainbow matching problem to a matching problem on a special hypergraph H.We then combine several existing techniques on matchings in uniform hypergraphs:Find an absorbing matching M in H;use a randomization process of Alon et al.(2012)to find an almost regular subgraph of H−V(M);find an almost perfect matching in H−V(M).To complete the process,we also need to prove a new result on matchings in 3-uniform hypergraphs,which can be viewed as a stability version of a result of Luczak and Mieczkowska(2014)and might be of independent interest.展开更多
文摘An edge coloring of hypergraph H is a function such that holds for any pair of intersecting edges . The minimum number of colors in edge colorings of H is called the chromatic index of H and is denoted by . Erdös, Faber and Lovász proposed a famous conjecture that holds for any loopless linear hypergraph H with n vertices. In this paper, we show that is true for gap-restricted hypergraphs. Our result extends a result of Alesandroni in 2021.
文摘Erdosa and Sós conjectured in 1963 that if G is a graph o ofof ordeq >1/2p(k - 1), then G contains every tree of size k. It is shown in this paper that the conjecture is true if the complement G of G contains no a copy of K3 as an induced subgraph of G.
基金Supported by National Natural Science Foundation of China
文摘Let Xn,n ≥ 1, be a sequence of independent random variables satisfying P(Xn = 0) = 1 - P(Xn = an) = 1 - 1/Pn, where an,n ≥ 1, is a sequence of real numbers, and Pn is the nth prime,set FN(x) = P (N Xn ≤ x). The authors investigate a conjecture of Erdos in probabilistic number theory and show that in order for the sequence FN to be weakly convergent, it is both sufficient and necessary that there exist three numbers X0 and X1 < X2 such that limsup(FN(X2) - FN(X1)) > 0 holds, and Lo = N→ ∞ lim FN(X0) exists. Moreover, the authors point out that they can also obtain the same result in the weakened case of lim inf P(Xn = 0) > 0.
基金supported by the National Key R and D Program of China(Grant No.2020YFA0713100)National Natural Science Foundation of China(Grant Nos.11871391,11622110 and 12125106)+1 种基金Fundamental Research Funds for the Central Universities,Anhui Initiative in Quantum Information Technologies(Grant No.AHY150200)National Science Foundation of USA(Grant No.DMS-1954134).
文摘Aharoni and Howard and,independently,Huang et al.(2012)proposed the following rainbow version of the Erd os matching conjecture:For positive integers n,k and m with n≥km,if each of the families F1,……,Fm⊆([n]k)has size more than max{(n k)−(n-m+1 k);(km-1 k)},then there exist pairwise disjoint subsets e1,……,em such that ei∈Fi for all i∈[m].We prove that there exists an absolute constant n0 such that this rainbow version holds for k=3 and n≥n_(0).We convert this rainbow matching problem to a matching problem on a special hypergraph H.We then combine several existing techniques on matchings in uniform hypergraphs:Find an absorbing matching M in H;use a randomization process of Alon et al.(2012)to find an almost regular subgraph of H−V(M);find an almost perfect matching in H−V(M).To complete the process,we also need to prove a new result on matchings in 3-uniform hypergraphs,which can be viewed as a stability version of a result of Luczak and Mieczkowska(2014)and might be of independent interest.
