Exact Results on the Number of Restricted Edge Colorings for Some Families of Linear Hypergraphs

Abstract

For k‐uniform hypergraphs F and H and an integer r≥2, let cr,F(H) denote the number of r‐colorings of the set of hyperedges of H with no monochromatic copy of F and let cr,F(n)=trueprefixmaxH∈scriptHncr,F(H), where the maximum is taken over the family Hn of all k‐uniform hypergraphs on n vertices. Moreover, let ex (n,F) be the usual extremal function, i.e., the maximum number of hyperedges of an n‐vertex k‐uniform hypergraph which contains no copy of F. Here, we consider the question for determining cr,F(n) for F being the k‐uniform expanded, complete graph Hℓ+1k or the k‐uniform Fan(k)‐hypergraph Fℓ+1k with core of size (ℓ+1), where ℓ≥k≥3, and we show cr,F(n)=r ex (n,F)for r=2,3 and n large enough. Moreover, for r=2 or r=3, for k‐uniform hypergraphs H on n vertices, the equality cr,F(H)=r ex (n,F) only holds if H is isomorphic to the ℓ‐partite, k‐uniform Turán hypergraph on n vertices, once n is large enough. On the other hand, we show that cr,F(n) is exponentially larger than r ex (n,F), if r≥4.

Edge-colorings of uniform hypergraphs avoiding monochromatic matchings

“…By Theorem 1.2, for k ∈ {2, 3}, we have H n,r,c(k,ℓ) = H n,r,ℓ−1 , while H n,r,c(k,ℓ) ̸ = H n,r,ℓ−1 for k ≥ 4, so that the lower bound in (2) is tight if and only if k ∈ {2, 3}. This phenomenon replicates, for forbidden r-matchings I ℓ and k ∈ {2, 3}, what has been observed in other classes of forbidden graphs (such as complete graphs [1,15], odd cycles [1] and some bipartite graphs [9]) and special classes of forbidden hypergraphs [11][12][13]. More generally, Lemma 2.1 in [1] for graphs and the work in [13] for arbitrary r-uniform hypergraphs implies that, given an r-uniform hypergraph F and k ∈ {2, 3}, we have c k,F ,r (n) ≤ k ex(n,F )+o(n r ) .…”

“…Let k ≥ 4 such that (ℓ − 1)k ≡ 1 (mod 3) and fix n 0 as in Eq. (11). Up to isomorphism, the maximum of c k,ℓ,r (H) over all r-uniform hypergraphs H with vertex set of size n ≥ n 0 and minimum vertex cover of size either c(k, ℓ) − 1 or c(k, ℓ) is achieved by H n,r,c(k,ℓ)−1 or H n,r,c(k,ℓ) .…”

“…Let k and ℓ be positive integers and fix n 0 as in (11). For n ≥ n 0 , if H is an n-vertex r-uniform hypergraph with minimum vertex cover of cardinality c ≤ r(ℓ − 1)k, then…”

Maximum number of sum-free colorings in finite abelian groups

“…One possible approach to our problem is to use Green's regularity lemma for abelian groups [8]. In various analogous contexts regularity lemmas have proven to be a suitable tool [1,14,12,13,11]. While this may work well here for groups with many subgroups such as F n p the technical difficulties are considerable for those lacking subgroups.…”

“…Related results. Problems analogous to the ones considered in this paper were investigated for many other discrete structures (see, e.g., [20,1,14,12,13,11,10]).…”