Let $A$ and $B$ be two Young tableaux, i. e. Young diagrams filled with the numbers $1$, $2$, ..., $n$ for some $n$ (not necessarily the same $n$). (They need not be semistandard.)

(a) (Etingof's Lectures on Representation Theory, proof of Lemma 4.40): If $A$ and $B$ share the same Young diagram, then there exist two entries which lie in the same row of $A$ and in the same column of $B$, unless we can make $A$ and $B$ equal by permuting some elements inside their rows in $A$ and permuting some elements inside their columns in $B$.

(b) (Etingof's Lectures on Representation Theory, proof of Lemma 4.41): If the partition corresponding to $A$ is lexicographically larger than that corresponding to $B$ - but both have the same sum -, then there exist two entries which lie in the same row of $A$ and in the same column of $B$.

(c) (Serganova's Representation Notes, lecture 6, Lemma 1.4): If the Young diagram of $B$ has only one less square than $A$ but does not result from $A$ by removing one square, then there exist two entries which lie in the same row of $A$ and in the same column of $B$, or two entries which lie in the same row of $B$ and in the same column of $A$.

I am rather new to Young tableaux, and I haven't looked into Fulton, Stanley or Knuth, but maybe someone can answer on the spot whether there is a more general statement behind these three results?

Oh, and since this fits so nicely: This paper gives a wonderful proof of the Littlewood-Richardson rule, even generalized to the product of a Schur function of a Young diagram with that of a skew Young diagram. Is there a reasonable generalization to the product of two skew Young diagrams?