How to prove this relation between Ramsey Numbers: $R(s,t)≤R(s,t−1)+R(s−1,t)-1$ for $s,t>2$ when $R(s,t-1)$ and $R(s-1,t)$ are even.
56010
Strengthening of a lemma in Ramsey Theory: $R(m, n) \le R(m-1, n) + R(m, n-1)-1$ if both numbers are even
136737
Proof by Induction for inequality, $\sum_{k=1}^nk^{-2}\lt2-(1/n)$
113083
How to prove a set equality?
134989
Prove that if the identity is written as the product of $r$ transpositions, then $r$ is an even number
110103
Prove that there are $0<c_1 < c_2<1$ s.t. $\frac{1}{g'(c_1)}+\frac{1}{g'(c_2)}=2$ . The same for $g'(c_1)g'(c_2)=1$.
37569
Prove that $\binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1} $
130543
If $a_n = \sum\limits_{r=0}^n \binom{n}{r} b_r$, prove $(-1)^n b_n = \sum\limits_{s=0}^n \binom{n}{s} (-1)^s a_s$
47799
Show that the number or $r$ combinations of $X$ which contain no consecutive integers is given by $\binom{n-r+1}{r}$
65395
Prove that if $\operatorname{rank}(T) = \operatorname{rank}(T^2)$ then $R(T) \cap N(T) = \{0\}$
51706
If $(k,l) = 1$, show that $(b_2k-b_1l, a_1l-a_2k) = 1$ for $a_1b_2-a_2b_1=1$
146867
Combinatorial Proof Of A Number Theory Theorem--Confusion
13026
Prove that if AB=I_m, then rk(A)=rk(B).
9522
Proving $k \binom{n}{k} = n \binom{n-1}{k-1}$
104903
Lower bound for Ramsey numbers: $R( n + 2 , 3 )>3n$
52095
Prove either $G=ST$ or |$G|\geq|S|+|T|$
19571
Combinatorial proof for $\sum_{k = 0}^n \binom {r+k} k = \binom {r + n + 1} n$
105985
Prove that all even integers $n \neq 2^k$ are expressible as a sum of consecutive positive integers