On the Tur\'an density of $\{1, 3\}$-Hypergraphs

In this paper, we consider the Tur\'an problems on $\{1,3\}$-hypergraphs. We prove that a $\{1, 3\}$-hypergraph is degenerate if and only if it's $H^{\{1, 3\}}_5$-colorable, where $H^{\{1, 3\}}_5$ is a hypergraph with vertex set $V=[5]$ and edge set $E=\{\{2\}, \{3\}, \{1, 2, 4\}, \{1, 3, 5\}, \{1, 4, 5\}\}.$ Using this result, we further prove that for any finite set $R$ of distinct positive integers, except the case $R=\{1, 2\}$, there always exist non-trivial degenerate $R$-graphs. We also compute the Tur\'an densities of some small $\{1,3\}$-hypergraphs.


Background
Turán theory is an important and active area in the extremal combinatorics. In 1941, Turán [10] determined the graph with maximum number of edges among all simple graphs on n vertices that doesn't contain the complete graph K ℓ as a sub-graph. For any ǫ > 0, the Turán density π(H) of a graph H is the least number α so that any large graph with edge density (α + ǫ) will always contain a sub-graph isomorphic to H. Erdős-Simonovits-Stone theorem [1,2] determined the Turán densities of all non-bipartite graphs. (The Turán density of any bipartite graph is always 0. Those are called the degenerate graphs. ) Turán problems on uniform hypergraphs have been actively studied for many decades. However, on non-uniform hypergraphs, these problems are rarely considered. Johnston and Lu [4] established the framework of the Turán theory for non-uniform hypergraphs. A hypergraph H = (V, E) consists of a vertex set V and an edge set E ⊆ 2 V . Here the edges of E could have different cardinalities. The set of all the cardinalities of edges in H is denoted by R(H), the set of edge types. In this paper, we will fix a finite set R of positive integers and consider all simple hypergraphs H with R(H) ⊆ R, which are called R-hypergraphs (or R-graphs, for short). We say a hypergraph is simple if there is at most one edge connecting any collection of vertices. A general hypergraph allows every edge to be a multi-set of vertices.
For example, {2}-graphs are just graphs and {r}-graphs are just r-uniform hypergraphs. An R-graph H on n vertices is denoted as H R n . We denote H r as the rth level hypergraph of H which consists of all edges of cardinality r of H. We denote K R n as the complete hypergraph on n vertices with edge set ∪ i∈R [n] i . We say H ′ is a sub-graph of H, denoted by H ′ ⊆ H, if there exists a 1-1 map f : V (H ′ ) → V (H) so that f (e) ∈ E(H) for any e ∈ E(H ′ ). A necessary condition for H ′ ⊆ H is R(H ′ ) ⊆ R(H). A chain C R is a special R-graph containing exactly one edge of each size such that any pair of these edges are comparable under inclusion relation.
To measure the edge density of a non-uniform hypergraph, we use the Lubell function, which is the expected number of edges in the hypergraph hit by a random full chain [4]. For a non-uniform hypergraph G on n vertices, the Lubell function of G is defined by Given a family of hypergraphs H with common set of edge types R, we say G is H-free if G doesn't contain any member of H as a sub-graph. Let π n (H) be the maximum edge density of any H-free R-graph on n vertices. The Turán density of H is defined to be: A hypergraph G := G R n is extremal with respect to the family H if G is H-free and h n (G) is maximized.
Lu and Johnston [4] proved that this limit always exists by a simple average argument of Katona-Nemetz-Simonovits theorem [6]. They completely classified the Turán densities of {1, 2}-graphs.
where H 2 ∈ H is the graph with all edges of cardinality 2. P 2k is a closed path of length 2k, and X (H 2 ) is the chromatic number of H 2 .
It is trivial that π(H) ≤ |R(H)| and it is easy to see that π(H) ≥ |R(H)|− 1, since we can take an (|R(H)| − 1)-complete hypergraph K |R(H)|−1 n without the appearance of H. We are interested in these R-graphs with the smallest Turán density.
What do the degenerate R-graphs look like? For the special case R = {r}, Erdős [3] showed that an r-uniform hypergraph H is degenerate if and only if it is r-partite, that is, a sub-graph of a blow-up of a single edge of cardinality r. As a natural extension of a single edge, the chain C R for any set R is degenerate. Thus every sub-graph of a blow-up of a chain is also degenerate. We say a degenerate R-graph is trivial if it is a sub-graph of a blow-up of the chain C R . For R = {1, 2}, by Theorem 1 all degenerate {1, 2}-graphs are trivial. However, a nontrivial degenerate {2, 3}-graph is found in [4]. It indicates that this question is more intrigue for other R-graphs.
In this paper, we will give a necessary and sufficient condition for the degenerate {1, 3}-graphs. Given  3 ). For any {1, 3}-graph H with π(H) ≤ α, it must be the case that π(H) = 1.
We also obtain the Turán densities of some 3-partite {1, 3}-graphs, the results are shown in Section 4.
The paper is organized as follows. In section 2 we introduce some notations and lemmas for non-uniform hypergraphs. In section 3 we will prove the Theorem 2, Theorem 4 and Corollary 1. In section 4, we determine the Turán densities of some 3-partite {1, 3}-graphs. In section 5, we prove Theorem 3.

Notation and lemmas
In this section, we introduce some notations and lemmas for R-graphs and then for the {1, 3}-graphs. We call an edge of cardinality i as an i-edge, for each i ∈ R. For convenience, we call a vertex that forms a 1-edge as "black vertex", otherwise, "white vertex". We use notations of form H • n to represent a hypergraph on n vertices that contains only one "black vertex", similarly, H •• n represents a hypergraph on n vertices that contains two "black vertices", and so on. To simplify our notations for {1, 3}-graphs, we use form of abc to denote the edge {a, b, c}.
For a fixed set R = {k 1 , k 2 , . . . , k r }, with (k 1 < k 2 < . . . < k r ), R-flag is an R-graph containing exactly one edge of each size. The chain C R is the special R-flag with the edge set E( For any R-flag L, we have π(L) = |R| − 1 (see [4]). Thus the chain C {1,3} is a degenerate {1, 3}-graph.
The following definitions and lemmas on non-uniform hypergraphs are generalized from uniform hypergraphs.
The blow-up operation does not change the Turán density.
A direct corollary of Theorem 5 is the following result.
It is easy to generalize the concepts of homomorphisms and H-coloring to general R-graphs. Note that, if there exists a homomorphism from G to H, then G is isomorphic to a sub-graph of a blow-up of H. Thus we have:

R-graphs with loops, blow-up, and Lagrangian
A loop edge is a multiset of vertices. Sometimes we need to enlarge the concept of R-graphs to R-graphs with loops. For example, consider a {1, 3}-graph H 1 with the edge set {x, xyy, yyy}. Here xyy is a loop edge with vertex x occurring once and vertex y twice. In general, a loop edge e = x m1 1 · · · x m l l consists of m 1 copies of vertex x 1 , m 2 copies of vertex x 2 , and so on. For a loop edge e = x m1 1 · · · x m l l , the cardinality of e is |e| = i m i . We also define a multinomial coefficient c e to be Definition 5. The polynomial form of an R-graph H with loops on n vertices, For any R-graph H (with possible loops), one can construct the family of H-colorable R-graph by blowing up H in a certain way. The Lagrangian of H is the maximum edge density of the H-colorable R-graphs that one can get in this way. This definition of Lagrangian is the same as the one in [5]; but differs from the classical Lagrangian for r-uniform hypergraphs such as in [8] by a constant multiplicative factor. This is not essential. This is a special case of more general Lagrangian of non-uniform hypergraphs introduced by Peng-Wu-Yao [9].
Construction A: Consider a {1, 3}-graph (with loops) H A on two vertices {x, y} with edges {x, xyy, yyy}. The polynomial form of H A is It can be shown that λ(H, #» x ) reaches the maximum 1 + √ 3 18 over the simplex 6 )n such that all 1-edges are in X (drawn by a black point), and all 3-edges are either formed by three vertices in Y or by one vertex in X plus two vertices in Y . In another words, We have Here It is easy to check 3}-graph G B on n vertices is generated by blowing-up H B as follows: set a vertex partition V (G B ) = A ∪ B ∪ C. All 1-edges are in A and B(drawn by black points), all 3-edges are formed by exactly one vertex in each partition. We have n.

Product of two R-graphs
Let's define the product of R-graphs (with loops): Definition 6. For any two general R-graphs H 1 and H 2 with vertices set V 1 and V 2 respectively, we define the product of H 1 and H 2 , which is denoted by consists of all products of e × σ f , where σ = (σ(1), . . . , σ(r)) takes over all permutations of [r]. For example, given e = {v 1 , . . . , v r } ∈ E(H 1 ), and Define a map f : Then we have Thus the map f takes edges in H to edges in H 1 × H 2 , it is a graph homomorphism. Therefore, H is (H 1 × H 2 )-colorable. We first prove the following lemma.

Remark 1. K ••
3 is not contained in G A whose edge density reaches 1 +  Denote E(G 3 ) as the set of all 3-edges of G. To forbidden K •• 3 , there is at most one black vertex in any 3-edges of G, thus we have We consider the 3-edges of G in edge set S 1 × S 2 . Define y as the average edge density of such 3-edges in G. Thus Note that there exists one vertex s 0 ∈ S such that |C(s 0 )| ≥ y × |S| 2 , where C(s 0 ) is the set of 3-edges that contain the black vertex s 0 . For any vertex u ∈ S, define W u := {v ∈ S|s 0 uv ∈ E(G)}.
We then have and To forbidden G • 4 , if s 0 uv, s 0 uk ∈ E(G), then uvk ∈ E(G). Since for each u ∈ S, there are |Wu| 2 pair of vertices each can form a 3-edge with u, we need to remove these edges in S 3 . Let N be the number of 3-edges in S 2 but not in G, by Cauchy-Schwarz inequality, we have Thus we have When x ≤ 2 5 , the above expression reaches the maximum value when y = 3 2 x 1−x ≤ 1. When x ≥ 2 5 , the above expression reaches the maximum value when y = 1. Thus we obtain By Lemma 3 and Lemma 4, the result follows.

Proof of Theorem 4
In this subsection, we will consider the non-degenerate {1, 3}-graphs. In particularly, we consider the non-degenerate 3-partite {1, 3}-graphs. A hypergraph is called 3-partite if its vertex set V can be partitioned into 3 different classes V 1 , V 2 , V 3 such that every edge intersects each class in exactly one vertex.
So far we know that the chain C {1,3} = {1, 123} is 3-partite and it is degenerate, while a slightly larger 3-partite {1, 3}-graph 18 since it's not contained in the G A . Now we are ready to prove Theorem 4 and Corollary 1.
, 3}-graph G on n vertices, let X be the set of all 1-edges in G, and Y ⊆ V (G) be the complement of X. On one hand, 18 . Thus, we have π(H) = π(K •• 3 ) = 1 +
. On one hand, consider an extremal K ••• 3 -free {1, 3}-graph G n . Let X be the set vertices of 1-edges in G n . Projecting all the vertices in X into a single vertex x and all the vertices not in X into a single vertex y, we get an {1, 3}graph (with loops) H C : where E(H c ) = {x, xxy, xyy, yyy}. This projection is a hypergraph homomorphism from G n to H C since G is K ••• 3 -free. Thus G is H c -colorable. In particular, we have .
On the other hand, any blow-up of H c does not contain the sub-graph K ••• 3 . The blow-up graph G C has the maximal edge density 1 + 2  , since they are not contained in G E and G F respectively(lim n→∞ h n (G E ) = lim n→∞ h n (G F )).
To calculate the upper bounds of π(H * 5 ), we need the following lemma.
To forbidden H •• 4 , for each pair of vertices (i, j) ∈ X 2 , if ijk and ijl are in E(G), neither kli nor klj can be contained in E(G). Thus for every pair of {i, j}, the number of 3-edges not shown in G is at least 2 dij 2 . Let M be the total number of 3-edges of form X 1 × Y 2 not shown in G, then by Cauchy-Schwarz inequality, we have Thus A simple calculation can show that h n (G) achieves maximum value at y = 1, which implies that for any positive integer n, any extremal {1, 3}-graph . The result follows.
Proof. On one hand, H * 5 is not contained in G E , then π(H * 5 ) ≥ 1 + -colorable, in this case the Turán density is 1.
To calculate the upper bounds of π(H * 6 ), we need the following lemma.
Lemma 11. Let R be a set of two distinct positive integers, R = {1, 2}. Then there exist non-trivial degenerate R-graphs.
Proof. By Corollary 3, for every positive integer k, one can take the suspension of H Lemma 12. Let R be a set of distinct positive integers with |R| ≥ 2 and 1 ∈ R. If there exist non-trivial degenerate R-graphs, then there exist non-trivial degenerate {1} ∪ R-graphs.
Proof. For each R stated in the lemma, let H be the non-trivial degenerate R-graph. Let H ′ be the disjoint union of H with a single 1-edge v ∈ H. Clearly, H ′ is not contained in a blow-up of chain C {1}∪R . We will prove that H ′ is also degenerate.
Let n be a positive integer and G = (V, E) be an extremal H ′ -free {1} ∪ Rgraph on n vertices. We have π n (H ′ ) = h n (G). Denote E i as the set of i-edges of G, for each i ∈ {1} ∪ R. For any 1-edge v ∈ E 1 , consider the sub-graph G v of G by removing all 1-edges (keep the vertices of these 1-edges in G v ). Then the vertex set V (G v ) = V , set of i-edges E i (G v ) = E i (G) for each i ∈ R. Then we have Observe that G v is an H-free R-graph on n vertices, so π n (H) ≥ h n (G v ). Then we have ≤ 1 + π n (H).
Proof of Theorem 3. Using the non-trivial degenerate R-graph for R stated in Lemma 11, then apply Lemma 12, we obtain non-trivial degenerate R-graphs for |R| = 3 and 1 ∈ R. Apply Corollary 3, we then obtain all other non-trivial degenerate R-graphs for |R| = 3. Repeatedly apply Lemma 12 and Corollary 3, we can obtain all R-graphs for |R| ≥ 4, the result follows.
We conjecture that for any set R, there exists an R-graph H R such that if G R is R-degenerate if and only if G R is H R -colorable. This conjecture is true for the case R = {r} with r ≥ 2 and R = {1, 2} and is confirmed for R = {1, 3} in this paper.