Problems

In an intersecting r-partite hypergraph, what is the smallest size of a vertex cover that does not contain any edge or side?

Let b_t(n) denote the minimum number of edges induced by any set of n / 2 vertices in the Turán graph on n vertices for K_t .If each set of ⌊ n/2 ⌋ vertices in a graph G of order n spans more than b_t(n) edges, then G contains a K_t .

Conjecture 4.2. Let h≥1,(a_{1},...,a_{h})\in \mathbb{C}^{h} , and P(x)\in \mathbb{C}[x] . Set I_h,P,n(x)=P(x)∏_i=1^n(1+a_1x^F_i+a_2x^F_i+1+… +a_hx^F_i+h-1).Regarding h, P as fixed, let c_{n}(p) denote the coefficient of x^{p} in…

There exists a constant K such that for every integer m, where m ≥ K, there exists an integer n such that χ_st'(C_m square C_n) = 6.

For an arbitrary planar graph G, is there a proper grid drawing of G in a grid of polynomial size?

Is it true that for sufficiently large n, indeed we can have min_i ∈[n]deg_B(i)deg_C(i)≤binomn-2k-2^2?

Does there exist a Hall function for the pairwise disjointness relation of the sets of a given family F?

It seems an interesting problem to characterize semisymmetric graphs with Wiener dimension 2.

Let 2≤ℓ<k and n ≠5. If G is an n-vertex k-chromatic ℓ -connected graph,then i(G)≤ i(G^*).

If A and B are finite sets of distinct vectors in R^n then show that the order of the set D(A, B) = d(a, b) : a ∈ A, b ∈ B is at least min|A|, |B|.

In addition, we conjecture that also (⌈n/2⌉ − k)-RCS is in P for any constant k.

(2) χ(G(2,8,6))=4?

(i) If T is a p-tree, then θ(T) is single-valued if p ≥ 6. (ii) If T is a p-tree, then for each p ≥ 7: θ(T)= llp-2 ifΔ(T)geqslant p-3, p-1 otherwise. .

For a given endofunction T, what is σ(d;T) ?

What is H(n;a_1,a_2,...,a_k) ? Is the upper bound given in Theorem 16 tight?

Every k-ary tangram T satisfies cut(T, S_k) ≤ c_k, for some finite constant c_k depending only on k.

Could it be true that for any finite dimensional \mathfrak{gl}{n} -module W there exists a polynomial p{W}(t)(p_{W}(t)=t ??) such that for all partitions \pi and \mu one has if a_{N \mu,W}^{N \pi}\ge p_{W}(N) , then a_{\mu,W}^{\pi}\ne 0 .

If A is [k]^(4) then this set is precisely A itself, but is it always the case that (for |A| > 1) we have |S(A)| ≥ |A|?

If 2 ≤ c ≤ d, then β_2(K_d^2× K_c^2)=d(c-1).

Given a p × q integer matrix M with p ≥ 2, if none of the differences between two rows of M is parallel to 1^{T} , then m(M,n)=(2+o(1))n/log_{p}n.